Unit 3 Trigonometric & Polar Functions

Unit 3: Trigonometric & Polar Functions 3.1 Periodic Phenomena PERIODIC PATTERNS: 3.1.A.1 A repeating pattern can be observed between two aspects of a context as the input values increase and the output values demonstrate a periodic relationship over equal-length intervals. - A periodic pattern occurs when two aspects of a context exhibit a repeating pattern over time. - To identify a periodic pattern, it is necessary to observe that as the input values increase, the values demonstrate a repeating pattern that occurs over equal-length intervals. Example: Let's consider a pendulum swinging back and forth. The position of the pendulum at any given time can be represented by an angle. As the pendulum swings, the angle it makes with the vertical line changes. If you measure the angle at regular intervals, you will notice that the angle values repeat after a certain interval of time, indicating a periodic pattern. 3.1.A.2 The graph of a periodic pattern can be constructed from the graph of a single cycle of the pattern. - To construct the graph of a periodic pattern, you can start with the graph of a single cycle. - A single cycle represents the pattern that repeats over and over in the periodic relationship. - By duplicating and aligning the single cycle horizontally, you can create the graph of the complete periodic pattern. Example: Consider the function y = sin(x), which represents a periodic pattern. The graph of a single cycle of the function is a smooth curve that starts at (0, 0), reaches a maximum at (π/2, 1), returns to zero at (π, 0), reaches a minimum at (3π/2, -1), and completes the cycle at (2π, 0). To graph the complete periodic pattern, you can duplicate this cycle and align the duplicates next to each other. KEY CHARACTERISTICS OF PERIODIC FUNCTIONS: 3.1.B.1 The period of the function is the smallest positive value k such that f(x + k) = f(x) for all x in the domain. As a result, the behavior of a periodic function is completely determined by any interval of width k. - The period of a periodic function is the smallest positive value, denoted as k, for which the function repeats itself. - In other words, if you add the period to any x-value in the domain, the corresponding function values remain the same. - The behavior of a periodic function is completely determined by any interval of width k. This means if you understand the function's behavior within a single period, you can predict its behavior in any other period. Example: Consider the function y = cos(2x). The period of this function can be found by setting 2x + k = 2x, where k represents the period. Solving this equation, we get k = π. Therefore, the function has a period of π, and its behavior repeats every π units of x. 3.1.B.2 The period can be estimated by examining successive equal-length output values and identifying where the pattern begins to repeat. - To estimate the period of a periodic function, observe the output values and identify where the pattern starts to repeat. - By examining successive equal-length output values, you can determine the length of the repeating pattern, which corresponds to the period of the function. Example: Given the function y = 3sin(4x), calculate the output values for several equally spaced x-values. If you notice that the pattern of the output values repeats after a certain number of steps, you have identified the period. For instance, if the output values repeat every 2 units, the period of the function is 2. 3.1.B.3 Periodic functions exhibit characteristics similar to those of other functions, such as intervals of increase and decrease, different concavities, and various rates of change. However, with periodic functions, all characteristics found in one period of the function will be present in every period of the function. - Periodic functions display characteristics similar to those found in other types of functions, such as intervals of increase and decrease, different concavities, and varying rates of change. - However, in periodic functions, these characteristics are repeated in every period of the function. The behavior observed within one period will be present in all subsequent periods. Example: Consider the function y = tan(x). This periodic function has intervals of increase and decrease, changes in concavity, and varying rates of change within one period. As you extend the graph beyond the initial period, you will observe that these characteristics repeat in each subsequent period. SAMPLE QUESTIONS: 1.3.A.1 Identify a periodic relationship between two aspects of a context. Answer: A periodic relationship can be observed in the rising and falling tides, where the water level demonstrates a repeating pattern over equal-length intervals. 1.5.B.2 Given the verbal representation of a periodic function, describe its key characteristics. Answer: The key characteristics of a periodic function can be determined by analyzing a single period of the function. These characteristics include intervals of increase and decrease, different concavities, and various rates of change, all of which are repeated in every period of the function. 3.1.A.1 Construct the graph of a periodic relationship based on a verbal representation. Answer: Consider the verbal representation of a periodic relationship, such as "the temperature oscillates between 20°C and 30°C every 24 hours." By plotting the temperature values over successive equal-length intervals on a graph, you can construct the graph of the periodic relationship. 3.1.B.2 Estimate the period of the function f(x) = 2sin(3x). Answer: To estimate the period of the function, evaluate f(x) for several equally spaced x-values. If you notice that the pattern of the output values repeats every π/3 units, you can estimate the period to be π/3. 3.1.B.3 Describe how periodic functions differ from other functions. Answer: Periodic functions exhibit characteristics found in other functions, such as intervals of increase and decrease, different concavities, and varying rates of change. However, unlike other functions, all of these characteristics are repeated in every period of the function. 3.2 Sin, Cosine, & Tangent ESSENTIAL KNOWLEDGE: 3.2.A.1 Standard Position of an Angle: - An angle is said to be in standard position when its vertex coincides with the origin (0,0) of the coordinate plane, and one of its rays coincides with the positive x-axis. - The other ray of the angle is called the terminal ray. - Positive angles are measured counterclockwise from the positive x-axis, while negative angles are measured clockwise. Example 1: Consider an angle in standard position with a terminal ray that makes a 60° angle with the positive x-axis in the counterclockwise direction. Example 2: Consider an angle in standard position with a terminal ray that makes a -45° angle with the positive x-axis in the clockwise direction. 3.2.A.2 Radian Measure of an Angle: - The radian measure of an angle in standard position is the ratio of the length of the arc subtended by the angle to the radius of the unit circle. - For a unit circle (a circle with a radius of 1), the radian measure is equal to the length of the subtended arc. Example 1: For a unit circle, an angle of 1 radian has an arc length equal to the radius of the circle. Example 2: For a unit circle, an angle of π/3 radians has an arc length equal to one-third of the circumference of the circle. 3.2.A.3 Sine of an Angle: - Given an angle in standard position and a unit circle, the sine of the angle is equal to the y-coordinate of the point where the terminal ray intersects the circle. - The sine is the ratio of the vertical displacement of the point from the x-axis to the distance between the origin and the point. Example 1: For an angle of 30° in standard position, the sine is equal to 0.5. This means that the y-coordinate of the point where the terminal ray intersects the unit circle is 0.5. Example 2: For an angle of 135° in standard position, the sine is equal to -0.707. This means that the y-coordinate of the point where the terminal ray intersects the unit circle is -0.707. 3.2.A.4 Cosine of an Angle: - Given an angle in standard position and a unit circle, the cosine of the angle is equal to the x-coordinate of the point where the terminal ray intersects the circle. - The cosine is the ratio of the horizontal displacement of the point from the y-axis to the distance between the origin and the point. Example 1: For an angle of 45° in standard position, the cosine is equal to 0.707. This means that the x-coordinate of the point where the terminal ray intersects the unit circle is 0.707. 2: For an angle of -60° in standard position, the cosine is equal to -0.5. This means that the x-coordinate of the point where the terminal ray intersects the unit circle is -0.5. 3.2.A.5 Tangent of an Angle: - The tangent of an angle in standard position is the slope of the terminal ray, if it exists. - The tangent is the ratio of the vertical displacement to the horizontal displacement over any interval. - The tangent of an angle can also be expressed as the ratio of the sine of the angle to the cosine of the angle. Example 1: For an angle of 60° in standard position, the tangent is equal to √3 or approximately 1.732. This means that the slope of the terminal ray is √3. Example 2: For an angle of -30° in standard position, the tangent is equal to -0.577. This means that the slope of the terminal ray is -0.577. Sample Question 1: Find the sine, cosine, and tangent of an angle of 45°. Solution: To find the sine, cosine, and tangent of 45°, we can use the unit circle. - Sine of 45°: The sine is equal to the y-coordinate of the point where the terminal ray intersects the unit circle. Since the angle is 45°, which corresponds to the coordinate (1/√2, 1/√2) on the unit circle, the sine of 45° is 1/√2 or approximately 0.707. - Cosine of 45°: The cosine is equal to the x-coordinate of the point where the terminal ray intersects the unit circle. For 45°, the coordinate is (1/√2, 1/√2), so the cosine of 45° is also 1/√2 or approximately 0.707. - Tangent of 45°: The tangent can be calculated as the ratio of the sine to the cosine. Therefore, the tangent of 45 (1/√2)/(1/√2), which simplifies to 1. Answer: Sine of 45° = 1/√2 or approximately 0.707 Cosine of 45° = 1/√2 or approximately 0.707 Tangent of 45° = 1 Sample Question 2: Determine the sine, cosine, and tangent of an angle of -120°. Solution: To find the sine, cosine, and tangent of -120°, we need to find the corresponding point on the unit circle for -120°. - Sine of -120°: The sine is equal to the y-coordinate of the point where the terminal ray intersects the unit circle. For -120°, the coordinate is (-√3/2, -1/2), so the sine of -120° is -1/2. - Cosine of -120°: The cosine is equal to the x-coordinate of the point where the terminal ray intersects the unit circle. The coordinate for -120° is (-√3/2, -1/2), so the cosine of -120° is -√3/2. - Tangent of -120°: The tangent can be calculated as the ratio of the sine to the cosine. Therefore, the tangent of -120° is (-1/2)/(-√3/2), which simplifies to √3/3 or approximately 0.577. Answer: Sine of -120° = -1/2 Cosine of -120° = -√3/2 Tangent of -120° = √3/3 or approximately 0.577 These examples and sample questions should help your students understand how to determine the sine, cosine, and tangent of an angle in standard position using the unit circle. Practice with more angles and encourage your students to visualize the unit circle to reinforce their understanding. 3.3 Sine & Cosine Function Values TOPIC 3.3: Sine and Cosine Function Values LEARNING OBJECTIVES: Essential Knowledge: 3.3.A Determine coordinates of points on a circle centered at the origin. - 3.3.A.1 Given an angle of measure θ in standard position and a circle with radius r centered at the origin, there is a point, P, where the terminal ray intersects the circle. The coordinates of point P are (r cosθ, r sinθ). - 3.3.A.2 The geometry of isosceles right and equilateral triangles, while attending to the signs of the values based on the quadrant of the angle, can be used to find exact values for the cosine and sine of angles that are multiples of π/4 and π/6 radians and whose terminal rays do not lie on an axis. NOTES: 3.3.A.1: Coordinates of Points on a Circle When we have an angle θ in standard position (where the initial side lies along the positive x-axis and the vertex is at the origin) and a circle with radius r centered at the origin, we can determine the coordinates of a point P where the terminal ray intersects the circle. The coordinates of point P are given by (r cosθ, r sinθ). Here, r represents the radius of the circle. The cosine of an angle is the x-coordinate of the point on the circle, and the sine of an angle is the y-coordinate of the point on the circle. Example: Consider a circle with radius 5 centered at the origin. Find the coordinates of the point where the terminal ray of an angle of 45 degrees intersects the circle. Solution: Using the formula (r cosθ, r sinθ), we have: Coordinates of P = (5 cos 45°, 5 sin 45°) Using the values from the unit circle or a calculator: cos 45° = √2/2 ≈ 0.7071 sin 45° = √2/2 ≈ 0.7071 Coordinates of P = (5 * 0.7071, 5 * 0.7071) Coordinates of P ≈ (3.5355, 3.5355) Therefore, the coordinates of the point where the terminal ray of an angle of 45 degrees intersects the circle are approximately (3.5355, 3.5355). 3.3.A.2: Exact Values for Cosine and Sine of Special Angles To find exact values for the cosine and sine of angles that are multiples of π/4 and π/6 radians, and whose terminal rays do not lie on an axis, we can use the geometry of isosceles right and equilateral triangles. Isosceles Right Triangle: An isosceles right triangle has two equal sides and one right angle. When one of the acute angles is θ, the sides of the triangle have lengths 1, 1, and √2. Using this triangle, we can determine the exact values for cosine and sine of π/4 radians (45°) and π/2 radians (90°). Cosine and Sine values for π/4 radians (45°): In an isosceles right triangle with acute angle θ = π/4 radians (45°): - The adjacent side is equal to the cosine of the angle. - The opposite side is equal to the sine of the angle. Using the triangle with sides 1, 1, and √2: cos (π/4) = adjacent side/hypotenuse = 1/√2 = √2/2 ≈ 0 Certainly! Apologies for the cutoff. Here's the continuation of the notes: Cosine and Sine values for π/4 radians (45°): In an isosceles right triangle with acute angle θ = π/4 radians (45°): - The adjacent side is equal to the cosine of the angle. - The opposite side is equal to the sine of the angle. Using the triangle with sides 1, 1, and √2: cos (π/4) = adjacent side/hypotenuse = 1/√2 = √2/2 ≈ 0.7071 sin (π/4) = opposite side/hypotenuse = 1/√2 = √2/2 ≈ 0.707 Therefore, the exact values for the cosine and sine of π/4 radians (45°) are both approximately 0.7071. Equilateral Triangle: An equilateral triangle has three equal sides and three equal angles of 60 degrees or π/3 radians. Using this triangle, we can determine the exact values for cosine and sine of π/6 radians (30°) π/3 radians (60°). Cosine and Sine values for π/6 radians (30°): In an equilateral triangle with angle θ = π/6 radians (30°): - The adjacent side is equal to the cosine of the angle. - The opposite side is equal to half the sine of the angle. Using the equilateral triangle with side lengths 1, 1, and 1: cos (π/6) = adjacent side/hypotenuse = 1/2 = 0.5 sin (π/6) = opposite side/hypotenuse = (1/2) * √3 = √3/2 ≈ 0.866 Therefore, the exact values for the cosine and sine of π/6 radians (30°) are 0.5 and approximately 0.866, respectively. Cosine and Sine values for π/3 radians (60°): In an equilateral triangle with angle θ = π/3 radians (60°): - The adjacent side is equal to the cosine of the angle. - The opposite side is equal to the sine of the angle. Using the equilateral triangle with side lengths 1, 1, and 1: cos (π/3) = adjacent side/hypotenuse = 1/2 = 0.5 sin (π/3) = opposite side/hypotenuse = √3/2 ≈ 0.866 Therefore, the exact values for the cosine and sine of π/3 radians (60°) are 0.5 and approximately 0.866, respectively. SAMPLE QUESTIONS: 1. Discover the coordinates of the point on a circle with a radius of 8 where the terminal ray of an angle of 150 degrees intersects the circle. Solution: Using the formula (r cosθ, r sinθ), we have: Coordinates of P = (8 cos 150°, 8 sin 150°) Using the values from the unit circle or a calculator: cos 150° = -√3/2 ≈ -0.866 sin 150° = 1/2 = 0.5 Coordinates of P = (8 * -0.866, 8 * 0.5) Coordinates of P ≈ (-6.928, 4) Therefore, the coordinates of the point where the terminal ray of an angle of 150 degrees intersects the circle are approximately (-6.928, 4). 2. Find out the exact value of sin(π/3). Solution: Utilizing the knowledge that sin(π/3) is the y-coordinate of a point on the unit circle for an angle of π/3 radians (60°), we can recall that sin(π/3) = √3/2 ≈ 0.866. Therefore, the exact value of sin(π/3) is √3/2 or approximately 0.866. 3. Determine the exact values of cos(π/4) and sin(π/4). Solution: Using the knowledge that cos(π/4) and sin(π/4) are the x-coordinate and y-coordinate, respectively, of a point on the unit circle for an angle of π/4 radians (45°), we can recall that cos(π/4) = sin(π/4) = √2/2 ≈ 0.7071. Therefore, the exact values of cos(π/4) and sin(π/4) are both √2/2 or approximately 0.7071. 3.4 Sine & Cosine Function Graphs Essential Knowledge: - 3.4.A.1: Given an angle of measure θ in standard position and a unit circle centered at the origin, there is a point, P, where the terminal ray intersects the circle. The sine function, f(θ) = sinθ, gives the y-coordinate, or vertical displacement from the x-axis, of point P. The domain of the sine function is all real numbers. 3.4.A.2: As the input values, or angles, of the sine function increase, the output values oscillate between -1 and 1, taking every value in between and tracking the vertical distance of points on the unit circle from the x-axis. - 3.4.A.3: Given an angle of measure θ in standard position and a unit circle centered at the origin, there is a point, P, where the terminal ray intersects the circle. The cosine function, f(θ) = cosθ, gives the x-coordinate, or horizontal displacement from the y-axis, of point P. The domain of the cosine function is all real numbers. - 3.4.A.4: As the input values, or angles, of the cosine function increase, the output values oscillate between -1 and 1, taking every value in between and tracking the horizontal distance of points on the unit circle from the y-axis. -----------------------------------------------------------------Detailed Notes: 1. Introduction to Sine and Cosine Functions: - Sine and cosine functions are fundamental trigonometric functions used to describe relationships between angles and sides of a right triangle. - These functions can also be represented graphically using the unit circle. 2. The Unit Circle: - The unit circle is a circle with a radius of 1 unit, centered at the origin (0, 0) on the Cartesian plane. When an angle θ in standard position is formed, the terminal ray intersects the unit circle at a point P(x, y), where x is the cosine value and y is the sine value. 3. Sine Function (f(θ) = sinθ): - The sine function represents the vertical displacement from the x-axis of a point P on the unit circle. - The domain of the sine function is all real numbers. - As the input values (angles) increase, the output values (sine) oscillate between -1 and 1, taking every value in between. - Example: For an angle of 30 degrees (π/6 radians), the sine value is 0.5, indicating that the point P on the unit circle is located halfway up from the x-axis. - Example Question: What is the sine of 45 degrees? Answer: The sine of 45 degrees is √2/2, indicating that the point P on the unit circle is located at (√2/2, √2/2). 4. Cosine Function (f(θ) = cosθ): - The cosine function represents the horizontal displacement from the y-axis of a point P on the unit circle. - The domain of the cosine function is all real numbers. - As the input values (angles) increase, the output values (cosine) oscillate between -1 and 1, taking every value in between. - Example: For an angle of 60 degrees (π/3 radians), the cosine value is 0.5, indicating that the point P on the unit circle is located halfway to the right from the y-axis. - Example Question: What is the cosine of 30 degrees? Answer: The cosine of 30 degrees is √3/2, indicating that the point P on the unit circle is located at (√3/2, 1/2). 5. Graphical Representations: - The sine and cosine functions can be graphed to visually represent their values for different angles. - The x-axis represents the angle (θ), and the y-axis represents the function values (sinθ or cosθ). The graphs of both functions repeat their values periodically due to the cyclic nature of angles. - The graphs of the sine and cosine functions are sinusoidal waves. 6. Characteristics of Sine Function Graph: - The graph of the sine function starts at the origin (0, 0) on the unit circle. - The amplitude of the sine function is 1, representing the maximum vertical displacement from the x-axis. - The period of the sine function is 360 degrees (or 2π radians), indicating the length of one complete cycle. - The sine function is an odd function, meaning it is symmetric about the origin (y-axis). - Example: The graph of the sine function crosses the x-axis at 0 degrees, 180 degrees, and 360 degrees. 7. Characteristics of Cosine Function Graph: - The graph of the cosine function starts at the maximum point (1, 0) on the unit circle. - The amplitude of the cosine function is 1, representing the maximum horizontal displacement from the y-axis. - The period of the cosine function is 360 degrees (or 2π radians), indicating the length of one complete cycle. - The cosine function is an even function, meaning it is symmetric about the y-axis. - Example: The graph of the cosine function reaches its maximum at 0 degrees, its minimum at 180 degrees, and returns to the maximum at 360 degrees. 8. Relationship between Sine and Cosine Functions: - The sine and cosine functions are closely related and can be expressed in terms of each other using trigonometric identities. - The sine function is equal to the cosine of the complementary angle, and vice versa. - Example: sin(θ) = cos(90 - θ) and cos(θ) = sin(90 - θ) ------------------------------------------------------------------ Sample Questions: 1. Find the sine and cosine values for an angle of 60 degrees. - Answer: The sine of 60 degrees is √3/2, and of 60 degrees is 1/2. 2. Draw the graph of the sine function for angles ranging from 0 to 360 degrees. - Answer: The graph of the sine function starts at the origin, crosses the x-axis at 0, 180, and 360 degrees, and reaches its maximum and minimum points at 90 and 270 degrees. 3. Determine the cosine of 120 degrees. - Answer: The cosine of 120 degrees is -1/2. 4. Sketch the graph of the cosine function for angles ranging from 0 to 360 degrees. - Answer: The graph of the cosine function starts at the maximum point (1, 0), reaches its minimum at 180 degrees, and returns to the maximum at 360 degrees. 5. Find the relationship between the sine and cosine functions for an angle of 45 degrees. - Answer: sin(45 degrees) = cos(45 degrees) = √2/2 3.5 Sinusoidal FUnction TOPIC 3.5: Sinusoidal Functions 3.5.A Identifying key characteristics of the sine and cosine functions. 3.5.A.1 A sinusoidal function is any function that involves additive and multiplicative transformations of f(θ) = sinθ. The sine and cosine functions are both sinusoidal functions, with cosθ = sin(θ + π/2). The sine function, represented as f(θ) = sinθ, depicts the ratio of the length of the side opposite an angle in a right triangle to the hypotenuse. Similarly, the cosine function, represented as g(θ) = cosθ, illustrates the ratio of the length of the adjacent side to the hypotenuse. Example: Now, let's consider the angles θ = 30° and θ = 60°. The values of sinθ and cosθ for these angles are as follows: - sin(30°) = 0.5 - sin(60°) = √3/2 - cos(30°) = √3/2 - cos(60°) = 0.5 3.5.A.2 The period and frequency of a sinusoidal function are reciprocals. The period of f(θ) = sinθ and g(θ) = cosθ is 2π, and the frequency is 1/2π. The period of a sinusoidal function is the duration of one complete cycle of the function. Meanwhile, the frequency is the number of cycles per unit length or time. Example: For the function f(θ) = sinθ, the period is 2π radians or 360 degrees, and the frequency is 1/2π cycles per radian or degree. Sample Question: What is the period and frequency of the function g(θ) = cosθ? Answer: The period is 2π radians or 360 degrees, and the frequency is 1/2π cycles per radian or degree. 3.5.A.3 The amplitude of a sinusoidal function is half the difference between its maximum and minimum values. The amplitude of f(θ) = sinθ and g(θ) = cosθ is 1. The amplitude signifies the maximum displacement or height of the sinusoidal function from its midline. Example: For the function f(θ) = sinθ, the maximum value is 1 and the minimum value is -1. Therefore, the amplitude is (1 - (-1))/2 = 1. Sample Question: Find the amplitude of the function g(θ) = cosθ. Answer: The amplitude is 1. 3.5.A.4 The midline of the graph of a sinusoidal function is determined by the average, or arithmetic mean, of the maximum and minimum values of the function. The midline of the graphs of y = sinθ and y = cosθ is y = 0. The midline represents the horizontal axis around which the sinusoidal function oscillates. Example: For the function y = sinθ, the maximum value is 1 and the minimum value is -1. Thus, the midline is (1 + (-1))/2 = 0. Sample Question: Determine the midline of the graph of y = cosθ. Answer: The midline is y = 0. 3.5.A.5 As input values increase, the graphs of sinusoidal functions oscillate between concave down and concave up. The graphs of sinusoidal functions demonstrate a repetitive pattern of oscillation as the input values increase or decrease. The functions alternate between concave down and concave up at different points in the cycle. Example: The graph of y = sinθ oscillates between concave up and concave down as θ increases. At certain points in the cycle, the graph is concave up, resembling a "U" shape. At other points, the graph is concave down, forming an inverted "U" shape. Sample Question: Describe the concavity of the graph of y = cosθ. Answer: The graph of y = cosθ is always concave down, resembling an inverted "U" shape, as θ increases or decreases. 3.5.A.6 The graph of y = sinθ exhibits rotational symmetry about the origin and is therefore an odd function. The graph of y = cosθ possesses reflective symmetry over the y-axis and is therefore an even function. Rotational symmetry refers to the property of a graph where it remains unaltered when rotated by a certain angle around a specific point. In the case of y = sinθ, the graph remains the same when rotated by 180 degrees around the origin. Reflective symmetry, on the other hand, refers to the property of a graph where it remains unaltered when reflected over a specific axis. In the case of y = cosθ, the graph remains the same when reflected over the y axis. Example: The graph of y = sinθ remains unchanged when rotated by 180 degrees around the origin. Conversely, the graph of y = cosθ remains the same when reflected over the y-axis. Sample Question: Determine if the function y = sinθ is odd or even. Answer: The function y = sinθ is an odd function. To sum up, sinusoidal functions like f(θ) = sinθ and g(θ) = cosθ possess key characteristics, including their additive and multiplicative transformations, period and frequency, amplitude, midline, oscillation between concave up and concave down, and symmetry properties. Understanding these characteristics is crucial for analyzing and graphing sinusoidal functions. 3.6 Sinusoidal Function Transformations Topic 3.6: Sinusoidal Function Transformations 3.6.A Understanding the Characteristics of Sinusoidal Functions 1. Sinusoidal functions can be written in the form f(θ) = asin(b(θ + c)) + d or g(θ) = acos(b(θ + c)) + d, where a, b, c, and d are real numbers and a ≠ 0. These functions are transformations of the sine and cosine functions. The cosine function is a phase shift of the sine function by π/2 units. - The general form of a sinusoidal function is f(θ) = asin(b(θ + c)) + d or g(θ) = acos(b(θ + c)) + d. The amplitude (|a|) represents the maximum distance the graph reaches from the midline. - The reciprocal of the period (1/|b|) represents the number of cycles the function completes in 2π units. The phase shift (c) determines the horizontal translation of the graph. - The vertical shift (d) determines the vertical translation of the graph. Let's take an example to understand these characteristics better. Consider the function f(θ) = 3sin(2(θ - π/4)) + 2. - The amplitude is |a| = 3, indicating that the graph reaches a maximum value of 3 units above and below the midline. - The reciprocal of the period is |b| = 2, indicating that the function completes two cycles in 2π units. - The phase shift is c = π/4, indicating that the graph is shifted π/4 units to the right. - The vertical shift is d = 2, indicating that the graph is shifted 2 units upward from the midline. 2. Additive and multiplicative transformations have the same effect on both sine and cosine functions. 2.1 Additive Transformation: g(θ) = sin(θ) + d - The additive transformation g(θ) = sin(θ) + d shifts the graph of the sine function f(θ) = sin(θ) upward or downward by d units. - The same transformation applies to the cosine function. To illustrate this, let's consider the function f(θ) = sin(θ). The graph of g(θ) = sin(θ) + 2 is obtained by shifting the graph of f upward by 2 units. Sample Question: Q1. Given the function f(θ) = sin(θ), write the equation for the graph obtained by shifting the graph upward by 3 units. A1. The equation for the graph obtained by shifting f upward by 3 units is g(θ) = sin(θ) + 3. 2.2 Multiplicative Transformation: g(θ) = asin(θ) - The multiplicative transformation g(θ) = asin(θ) scales the graph of the sine function f(θ) = sin(θ) vertically by a factor of |a|. - The amplitude of the function g is |a| times the amplitude of f. - The same transformation applies to the cosine function. Let's consider the function f(θ) = sin(θ). The graph of g(θ) = 2sin(θ) is obtained by vertically stretching the graph of f by a factor of 2. Sample Question: Q2. Given the function f(θ) = sin(θ), write the equation for the graph obtained by vertically compressing the graph by a factor of 0.5. A2. The equation for the graph obtained by vertically compressing f by a factor of 0.5 is g(θ) = 0.5sin(θ). 2.3 The same transformations apply to the cosine function. 3. Shifts in the Graph: g(θ) = sin(θ + c) - The additive transformation g(θ) = sin(θ + c) shifts the graph of the sine function f(θ) = sin(θ) horizontally to the left or right by c units. - The same transformation applies to the cosine function. 4. Stretching and Compressing the Graph: g(θ) = sin(bθ) - The multiplicative transformation g(θ) = sin(bθ) scales the graph of the sine function f(θ) = sin(θ) horizontally by a factor of 1/|b|. - The period of the function g is 2π/|b|, which is the reciprocal of the frequency. - The same transformation applies to the cosine function. Let's consider the function f(θ) = sin(θ). The graph of g(θ) = sin(2θ) is obtained by horizontally compressing the graph of f by a factor of 2. Sample Question: Q3. Given the function f(θ) = sin(θ), write the equation for the graph obtained by horizontally stretching the graph by a factor of 3. A3. The equation for the graph obtained by horizontally stretching f by a factor of 3 is g(θ) = sin(θ/3). 5. Summary of Characteristics: y = asin(b(θ + c)) + - The amplitude of the sinusoidal function y = asin(b(θ + c)) + d is |a|. - The period of the function is 2π/|b|. - The midline is shifted vertically by d units from y = 0. - The phase shift is -c units, indicating a horizontal translation to the left or right. - The same transformations apply to the cosine function. 3.7 Sinusoidal Function Context & Data Modeling LEARNING OBJECTIVES: 3.7.A Construct sinusoidal function models of periodic phenomena by estimating key values. ESSENTIAL KNOWLEDGE: 3.7.A.1 The smallest interval of input values over which the maximum or minimum output values start to repeat can be used to determine or estimate the period and frequency for a sinusoidal function model. Explanation: In a sinusoidal function, the period refers to the length of one complete cycle of the function. It is the interval over which the values of the function start to repeat. To determine or estimate the period, we can observe the input values for which the maximum or minimum output values occur first and then repeat. For example, let's consider the function f(x) = sin(x). The period of this function is 2π, as the maximum and minimum values of the sine function repeat every 2π units. Sample Question: 1. Consider the function f(x) = cos(2x). What is the period of this function? Answer: The coefficient of x in the function is 2, which means the period is obtained by dividing 2π by the coefficient. In this case, the period is 2π/2 = π. 3.7.A.2 The maximum and minimum output values can be used to determine or estimate the amplitude and vertical shift for a sinusoidal function model. Explanation: The amplitude of a sinusoidal function refers to half the difference between the maximum and minimum output values. It represents the maximum displacement of the function from its central position. The vertical shift represents any upward or downward shift of the entire function. By analyzing the maximum and minimum values of the function, we can determine or estimate the amplitude and vertical shift. Sample Question: 2. Consider the function g(x) = 3sin(x) + 2. What is the amplitude and vertical shift of this function? Answer: The amplitude is half the difference between the maximum and minimum values of sine function, which is 3/2. The vertical shift is given by the constant term in the function, which is 2. 3.7.A.3 An actual pair of input-output values can be compared to pairs of input-output values produced by a sinusoidal function model to determine or estimate a phase shift for the model. Explanation: The phase shift of a sinusoidal function represents the horizontal shift of its graph compared to the standard form of the function y = A*sin(B(x - C)) + D. To determine or estimate the phase shift, we can compare an actual pair of input-output values from the given data with the corresponding pairs of input-output values produced by the sinusoidal function model. Sample Question: 3. The graph of a sinusoidal function crosses the x-axis at x = π/4. What is the phase shift of the function? Answer: The phase shift can be determined by comparing the given x-coordinate with the standard form of the function. In the standard form, the phase shift is represented by the value of C. So, in this case, the phase shift is π/4. 3.7.A.4 Technology can be used to find an appropriate sinusoidal function model for a data set. Explanation: Technology, such as graphing calculators or mathematical software, can be utilized to fit a sinusoidal function model to a given data set. By plotting the data points and using curve-fitting techniques, the technology can provide the equation of a sinusoidal function that best represents the data. Sample Question: 4. Given a set of data points, use technology to find an appropriate sinusoidal function model that fits the data: (1, 3), (2, 5), (3, 7), (4, 5). Answer: Using graphing software, plot the data points and use curve-fitting tools to find the best sinusoidal function model. For example, the software may provide the equation f(x) = 2sin(x) + 4 that fits the given data. 3.7.A.5 Sinusoidal functions that model a dataset are frequently only useful over their contextual domain and can be used to predict values of the dependent variable from a value of the independent variable. Explanation: When using a sinusoidal function to model a dataset, it is important to consider the contextual domain over which the function is valid. Sinusoidal functions are periodic, and their predictions for dependent variables beyond the contextual domain may not be accurate. However, within the contextual domain, sinusoidal functions can be utilized to predict values of the dependent variable based on a given value of the independent variable. Sample Question: 5. A sinusoidal function models the temperature of a location over a 24-hour period. Can this function accurately predict the temperature after 72 hours? Answer: No, a sinusoidal function with a 24-hour period is only valid within the 24-hour interval. Predictions beyond the 24-hour period may not be accurate, as the function assumes that the pattern within one day repeats itself. Therefore, the function may not accurately predict the temperature after 72 hours. 3.8 The Tangent Function 3.8.A Construct representations of the tangent function using the unit circle. 3.8.A.1 Given an angle of measure θ in standard position and a unit circle centered at the origin, there is a point, P , where the terminal ray intersects the circle. The tangent function, f(θ) = tanθ, gives the slope of the terminal ray. To construct representations of the tangent function, we use the unit circle and draw a horizontal line from the point P where the terminal ray intersects the circle to the y-axis. The point where the line intersects the y-axis represents the value of the tangent function for that particular angle. For example, if we have an angle of 30 degrees in standard position, we draw a unit circle and place the terminal ray at 30 degrees. The point P where the terminal ray intersects the circle has coordinates (cos(30), sin(30)) = (√3/2, 1/2). We then draw a horizontal line from P to the y-axis, which intersects the y-axis at √3/3. This value is the tangent of 30 degrees, so we write f(30) = tan(30) = √3/3. 3.8.A.2 Because the slope of the terminal ray is the ratio of the change in the y-values to the change in the x values between any two points on the ray, the tangent function is also the ratio of the sine function to the cosine function. Therefore, tanθ = sinθ, where cosθ≠0. This means that we can also find the value of the tangent function by dividing the value of the sine function by the value of the cosine function for that particular angle. For example, if we have an angle of 45 degrees in standard position, we draw a unit circle and place the terminal ray at 45 degrees. The point P where the terminal ray intersects the circle has coordinates (cos(45), sin(45)) = (√2/2, √2/2). The value of the sine function for 45 degrees is √2/2 and the value of the cosine function is also √2/2. Therefore, f(45) = tan(45) = sin(45)/cos(45) = (√2/2)/(√2/2) = 1. Sample question: Given an angle of 60 degrees, construct a representation of the tangent function using the unit circle. Solution: We draw a unit circle and place the terminal ray at 60 degrees. The point P where the terminal ray intersects the circle has coordinates (cos(60), sin(60)) = (1/2, √3/2). We then draw a horizontal line from P to the y axis, which intersects the y-axis at √3. This value is the tangent of 60 degrees, so we write f(60) = tan(60) = √3. 3.8.B Describe key characteristics of the tangent function. 3.8.B.1 Because the slope values of the terminal ray repeats every one-half revolution of the circle, the tangent function has a period of π. This means that the graph of the tangent function repeats every π radians, or every 180 degrees. 3.8.B.2 The tangent function demonstrates periodic asymptotic behavior at input values θ = π + kπ, for integer values of k, 2 because cosθ = 0 at those values. This means that the graph of the tangent function approaches infinity or negative infinity at these input values, as the denominator of the ratio (cosθ) becomes zero. 3.8.B.3 The tangent function increases and its graph changes from concave down to concave up between consecutive asymptotes. This means that the graph of the tangent function becomes steeper as we approach an asymptote and flips direction at each asymptote. Sample question: Describe the key characteristics of the tangent function. Solution: The tangent function has a period of π, meaning that the graph repeats every π radians, or every 180 degrees. The tangent function also demonstrates periodic asymptotic behavior at input values θ = π + kπ, for integer values of k, 2 because cosθ = 0 at those values, causing the graph to approach infinity or negative infinity. The tangent function increases and its graph changes from concave down to concave up between consecutive asymptotes, making the graph steeper as we approach an asymptote and flipping direction at each asymptote. 3.8.C Describe additive and multiplicative transformations involving the tangent function. 3.8.C.1 The graph of the additive transformation g(θ) = tanθ + d of the tangent function f(θ) = tanθ is a vertical translation of the graph of f and the line containing its inflection points by d units. This means that the graph of the tangent function shifts vertically by the value of d. 3.8.C.2 The graph of the additive transformation g(θ) = tan(θ + c) of the tangent function f(θ) = tanθ is a horizontal translation, or phase shift, of the graph of f by −c units. This means that the graph of the tangent function shifts horizontally by the value of c. 3.8.C.3 The graph of the multiplicative transformation g(θ) = a tanθ of the tangent function f(θ) = tanθ is a vertical dilation of the graph of f by a factor of a. If a < 0, the transformation involves a reflection over the x-axis. This means that the graph of the tangent function stretches or shrinks vertically by a factor of a. If a is negative, the transformation also involves a reflection over the x-axis. 3.8.C.4 The graph of the multiplicative transformation g(θ) = tan(bθ) of the tangent function f(θ) = tanθ is a horizontal dilation of the graph of f and differs in period by a factor of b1. If b < 0, the transformation involves a reflection over the y-axis. This means that the graph of the tangent function stretches or shrinks horizontally by a factor of b and the period of the function changes. If b is negative, the transformation also involves a reflection over the y-axis. 3.8.C.5 The graph of y = f(θ) = a tan(b(θ + c)) + d is a vertical dilation of the graph of y = tanθ by a factor of a, has a period of 1 π units, is a vertical shift of the line containing the inflection points of the graph of y = tanθ by d units, and is a phase shift of −c units. This means that the graph of the tangent function stretches or shrinks vertically by a factor of a and shifts vertically by the value of d. The period of the function is 1 π units and the graph shifts horizontally by the value of c. Sample question: Given f(θ) = tanθ, describe the transformation represented by g(θ) = 3tan(2θ - π/4) + 2. Solution: The multiplicative transformation tan(2θ - π/4) stretches the graph horizontally by a factor of 2 and shifts it horizontally to the right by π/8. The period of the function becomes 1 π/2 units. The additive transformation 3tan(2θ - π/4) vertically stretches the graph by a factor of 3 and shifts it vertically up by 2. Therefore, the graph represented by g(θ) = 3tan(2θ - π/4) + 2 is vertically stretched by a factor of 3, horizontally stretched by a factor of 2, shifts horizontally to the right by π/8, and shifts vertically up by 2. The period of the function becomes 1 π/2 units. 3.9 Inverse Trigonometric Functions 3.9.A.1 For inverse trigonometric functions, the input and output values are switched from their corresponding trigonometric functions, so the output value of an inverse trigonometric function is often interpreted as an angle measure and the input is a value in the range of the corresponding trigonometric function. Example: Let's consider the sine function. The sine function takes an angle measure as input and gives the ratio of the opposite side to the hypotenuse as output. The inverse sine function takes a ratio of the opposite side to the hypotenuse as input and gives an angle measure as output. Question: Find the value of angle θ if sin(θ) = 0.5. Solution: Using the inverse sine function, we have sin^(-1)(0.5) = θ. In this case, θ is equal to 30 degrees or π/6 radians. 3.9.A.2 The inverse trigonometric functions are called arcsine, arccosine, and arctangent (also represented as sin^(-1)x, cos^(-1)x, and tan^(-1)x). Because the corresponding trigonometric functions are periodic, they are only invertible if they have restricted domains. Example: Let's consider the cosine function. The cosine function takes an angle measure as input and gives the ratio of the adjacent side to the hypotenuse as output. Since the cosine function is periodic with a period of 2π, it is not invertible over its entire domain. To make it invertible, we need to restrict its domain to a specific interval. Question: Find the value of angle θ if cos(θ) = 0.5, with a domain restriction of [0, π]. Solution: Using the inverse cosine function with the restricted domain, we have cos^(-1)(0.5) = θ. In this case, θ is equal to 60 degrees or π/3 radians. 3.9.A.3 In order to define their respective inverse functions, the domain of the sine function is restricted to [- π/2, π/2], the cosine function to [0,π], and the tangent function to (-π/2, π/2). Example: Let's consider the tangent function. The tangent function takes an angle measure as input and gives the ratio of the opposite side to the adjacent side as output. To make the tangent function invertible, we need to restrict its domain to a specific interval. Question: Find the value of angle θ if tan(θ) = 1.5, with a domain restriction of (-π/2, π/2). Solution: Using the inverse tangent function with the restricted domain, we have tan^(-1)(1.5) = θ. In this case, θ is equal to approximately 56.31 degrees or 1.03 radians. Sample Question: 1. Find the value of angle θ if cos(θ) = -0.6, with a domain restriction of [0,π]. Solution: Using the inverse cosine function with the restricted domain, we have cos^(-1)(-0.6) = θ. In this case, θ is equal to approximately 131.81 degrees or 2.30 radians. Explanation: The inverse cosine function gives the principal value of the angle between 0 and π which has a cosine value of -0.6. Overall, the inverse trigonometric functions allow us to find angle measures given the ratios of sides or vice versa. It is important to understand the restricted domains and the difference between the trigonometric and inverse trigonometric functions. 3.10 Trigonometric Equations & Inequalities 3.10.A Solve equations and inequalities involving trigonometric functions. 3.10.A.1 Inverse trigonometric functions are useful in solving equations and inequalities involving trigonometric functions, but solutions may need to be modified due to domain restrictions. When dealing with trigonometric equations, we can use inverse trigonometric functions to help solve for angles. However, when using inverse trig functions, we must be aware of the limited domain of the inverse function. For example, the inverse sine function can only give us angles between -π/2 and π/2. If our equation or inequality includes angles outside of this domain, we must modify our solution. Example 1: Solve for x in the equation sin x = 0.5 We can use the inverse sine function to solve for x: sin⁻¹(0.5) = 30°. However, we must also consider that sin is positive in both the first and second quadrants, so our answer must be modified to include the angle in the second quadrant: x = 180°-30° or x = 150°. 3.10.A.2 Because trigonometric functions are periodic, there are often infinite solutions to trigonometric equations. Since trigonometric functions repeat their values every 2π radians, there may be multiple solutions to a trigonometric equation. We can use the general solution format to express all of the possible solutions. For example, for sin x = 1, we know that x must equal π/2 + 2kπ, where k is an integer that specifies which cycle of the sine function we are looking at. Example 2: Solve for x in the equation cos x = -0.5 We can use the inverse cosine function to find one solution: cos⁻¹(-0.5) = 120°. However, we also know that cosine is negative in the second and third quadrants, so there must be two other solutions. We can use the general solution format: x = 120° + 360°k or x = 240° + 360°k, where k is an integer. 3.10.A.3In trigonometric equations and inequalities arising from a contextual scenario, there is often a domain restriction that can be implied from the context, which limits the number of solutions. When working with contextual problems, we may be able to understand the domain of the trigonometric function based on the context given. For example, if we are working with angles that represent the height of an object, the domain must be limited to angles between 0 and π/2. Example 3: A ladder is resting against a wall, forming an angle of 60° with the ground. If the ladder is 20 feet long, how high does it reach on the wall? We know that the angle formed by the ladder and the ground is 60°, so we can use the sine function to find the height of the ladder on the wall: sin 60° = opposite/20. Solving for the opposite side, we get a height of 17.3 feet. However, we know that the height must be positive, so we must restrict our domain to 0° < θ < 90°. Sample questions: 1. Solve for x in the equation cos x = 0.8. Give your answer in degrees. Answer: x = 36.9° or x = 323.1° Explanation: The inverse cosine function gives us the solution cos⁻¹(0.8) = 36.9°. However, since cosine is positive in both the first and fourth quadrants, we must also consider the angle x = 360° - 36.9° = 323.1°. 2. Solve for x in the inequality 2sin x > 1, where 0° ≤ x ≤ 360°. Answer: x ∈ (30°, 150°) U (210°, 330°) Explanation: We can start by dividing both sides of the inequality by 2: sin x > 0.5. Using the inverse sine function, we find the solutions sin⁻¹(0.5) = 30° and 150°. However, we must also consider that sine is positive in both the first and second quadrants, so we can include the angles 30° + 360°k and 150° + 360°k. Finally, we must remove any angles that do not satisfy the original inequality, resulting in the solution x ∈ (30°, 150°) U (210°, 330°). 3. A Ferris wheel with a diameter of 240 feet makes one complete revolution in 3 minutes. At what height above the ground is a rider after 1.5 minutes? Answer: 120 feet Explanation: We can start by finding the period of the sine function that represents the height of the rider on the Ferris wheel: T = 3 minutes. Since the diameter of the Ferris wheel is 240 feet, the amplitude of the sine function is 120 feet. The starting point of the ride is a height of 120 feet, so we can write a sine function using these parameters: h(θ) = 120 sin(θπ/180 - π/2). After 1.5 minutes, the angle traversed is θ = 90°, so we can evaluate the sine function: h(90°) = 120 feet. 3.11 The Secant, Cosecant, and Cotangent Functions 3.11.A.1 The secant function, f(θ) = secθ, is one of the trigonometric functions that is defined as the reciprocal of the cosine function, where cosθ is not equal to zero. The secant function can also be defined as the ratio of the hypotenuse to the adjacent side of a right triangle. For instance, if a right triangle has an angle of θ and an adjacent side of length 2 with a hypotenuse of length 5, then we can determine the value of secθ as follows: secθ = hypotenuse / adjacent = 5 / 2 The secant function has vertical asymptotes where cosθ equals zero, which occurs when the argument (the value inside the trigonometric function) of secθ equals to (2n + 1)π/2, where n is an integer. These vertical asymptotes divide the real line into intervals on which the function is either positive or negative. The range of the secant function is (-∞,-1] u [1,∞). Example Question: Evaluate the value of sec(π/6). Solution: We know that cos(π/6) is equal to √3/2, so sec(π/6) is equal to 1/cos(π/6), which is equal to 2/√3. 3.11.A.2 The cosecant function, f(θ) = cscθ, is another trigonometric function that is defined as the reciprocal of the sine function, where sinθ is not equal to zero. The cosecant function can also be defined as the ratio of the hypotenuse to the opposite side of a right triangle. For example, if a right triangle has an angle of θ and an opposite side of length 3 with a hypotenuse of length 5, then we can find the value of cscθ as follows: cscθ = hypotenuse / opposite = 5 / 3 The cosecant function has vertical asymptotes where sinθ equals zero, which occurs when the argument of cscθ equals to nπ, where n is an integer. These vertical asymptotes divide the real line into intervals on which the function is either positive or negative. The range of the cosecant function is (-∞,-1] u [1,∞). Example Question: Evaluate the value of csc(3π/4). Solution: We know that sin(3π/4) is equal to √2/2, so csc(3π/4) is equal to 1/sin(3π/4), which is equal to √2. 3.11.A.4 The cotangent function, f(θ) = cotθ, is the reciprocal of the tangent function, where tanθ is not equal to zero. The cotangent function can also be defined as the ratio of the adjacent side to the opposite side of a right triangle. For instance, if a right triangle has an angle of θ and an adjacent side of length 4 and an opposite side of length 3, then we can determine the value of cotθ as follows: cotθ = adjacent / opposite = 4 / 3 The cotangent function has vertical asymptotes where tanθ equals zero, which occurs when the argument of cotθ equals to nπ, where n is an integer. These vertical asymptotes divide the real line into intervals on which the function is either positive or negative. Furthermore, the cotangent function is also decreasing between consecutive asymptotes. Example Question: Evaluate the value of cot(7π/4).