Trigonometry Study Guide

Trigonometry Study Guide Ratios in Right Triangles ○ Hypotenuse: the side opposite the right angle. The hypotenuse, as noted below, is the longest side of a right triangle ○ The other two sides of a right triangle are the opposite and adjacent sides. These sides are labeled in relation to an angle ○ The opposite side of a triangle is the side across from a given angle ○ The adjacent side of a triangle is the non-hypotenuse side that is located next to a given angle Ratios in Right Triangles (continued) ○ Congruent: Two triangles are congruent if they have the exact same size and shape - triangles are congruent if their corresponding angles are equal, and their corresponding sides have the same lengths ○ If two right triangles share an acute angle measure, they are similar by angleangle similarity. ○ The ratios of corresponding side lengths within the triangles are equal. The ratio of the side lengths of a right triangle relies on one acute angle measure. ○ The Pythagorean theorem can be used to find any missing side length of a right triangle as long as you know the other two lengths. However, now we can relate angle measures to the right triangle side lengths. ○ This allows us to find both missing side lengths when we only know one length and an acute angle measure. The acute angle measures in a right triangle can be found based on any two side lengths. Introduction to the Trigonometric Ratios ● To help using the ratios - remember: SOH CAH TOA ○ SIN = OPPOSITE / HYPOTENUSE ○ COS = ADJACENT / HYPOTENUSE ○ TAN = OPPOSITE / ADJACENT Solving for a Side Within a Right Triangle Using the Trigonometric Ratios ● 1: Determine which trigonometric ratio to use. ● 2: Create an equation using the trig ratio sine and then solve for the unknown . Solving for a Side Within a Right Triangle Using the Trigonometric Ratios (continued) ● Inverse Trig Functions ○ Inverse sine (sin ^−1) does the opposite of the sine. ○ Inverse cosine (cos ^−1) does the opposite of cosine ○ Inverse tangent tan (tan ^−1) does the opposite of tan ■ Sin ^-1 (x) is also called arcsin(x) ■ 1/ sin x is also called csc(x) ○ EXAMPLE : Sine and Cosine of Complementary Angles ● Pythagorean Trig identity: used to figure out the value of (mostly) the hypotenuse in a right triangle. A and b sides are the two "non-hypotenuse" sides of the triangle (Opposite and Adjacent sides). ○ a^2 + b^2 = c^2 ● These triangles can be 45-45-90 triangles or 30-60-90 triangles. Quadrants ● REMEMBER: “ALL STUDENTS TAKE CALCULUS” Modeling with Right Triangles ● Solving for a side ○ Given the measure of angle B = 40 degrees, and the length of the hypotenuse, and find the side opposite to angle B, the trigonometric ratio that contains both of these sides is the sine: ○ CALCULATE: ■ sin(< B) = AC / AB ■ sin (40 degrees) = AC / 7 , < B = 40 degrees, AB = 7 ■ 7 x sin(40 degrees) = AC ■ = 4.5 ● Solving for an angle ○ Trigonometry can also be used to find missing angle measures. For example, find the measure of ∠A in this triangle: ■ We are given the length of the side adjacent to the missing angle, and the length of the hypotenuse. The trigonometric ratio that contains both of those sides is the cosine: ■ CALCULATE: ● cos(< A) = AC / AB ● cos(< A) = 6 / 8 ● < A = cos^-1 (6/8) ● = 41.41 degrees The Reciprocal Trigonometric Ratios ● Non-Right Triangles & Trigonometry ● Law of Sines: the ratio of each side of a plane triangle to the sine of the opposite angle is the same for all three sides and angles ● Law of Cosines: the square of a side of a plane triangle equals the sum of the squares of the remaining sides minus twice the product of those sides and the cosine of the angle between them Unit Circle ○ The unit circle definition allows us to extend the domain of sine and cosine to all real numbers ○ The process for determining the sine/cosine of any angle θ is as follows: ■ Starting from (1,0), move along the unit circle in the counterclockwise direction until the angle that is formed between your position, the origin, and the positive x-axis is equal to θ ■ sin(θ) is equal to the y-coordinate of your point, and cos⁡(θ) is equal to the x-coordinate. Radians ○ Radians are alternate units used to measure angles in trigonometry. ○ A radian is the measure of an angle subtended at the center of a circle by an arc whose length is equal to the radius of that circle ○ Converting from Radians to Degrees & converting from Degrees to Radians The Pythagorean Identity ○ The Pythagorean identity tells us that no matter what the value of θ is, sin²θ+cos²θ is equal to 1. We can prove this identity using the Pythagorean theorem in the unit circle with x²+y²=1. Trigonometric Values of Special Angles ○ Special Angles; The values of the six trigonometric ratios cannot be calculated exactly for most angles. Nor can the exact value of an angle generally be found given the value of one of the ratios. ● Graphs of sin(x), cos(x), and tan(x) Trigonometric Values of Special Angles (continued) ○ The graph of y = sin x: The graph of the sine function is a nice, continuous wave that rolls along gently and keeps repeating itself. The domain, or x-values, of the sine function, includes all angles in degrees or all real numbers in radians, so the curve has no breaks or holes. ○ y = sin x and y = cos x look similar; in fact, the main difference is that the sine graph starts at (0,0) and the cosine at (0,1). ○ Tangent graphs ■ The graph of y = tan x is an odd one - mainly down to the nature of the tangent function. Returning to SOH CAH TOA - our trig functions, with tan x being opposite/adjacent, you can see that: ■ Tan 0 = 0, because the opposite side would have zero length regardless of the length of the adjacent side. ■ Tan 90 is not possible, because a right triangle can not have two right angles! As the angle approaches 90 degrees, our opposite side would approach infinity. ■ This means that the graph of y = tan x crosses the x-axis at 0, and has an asymptote at 90. This graph repeats after every 180 degrees, instead of every 360 ■ At x = 0 degrees, sin x = 0 and cos x = 1. Tan x must be 0 (0 / 1) Trigonometric Values of Special Angles (continued) ■ At x = 90 degrees, sin x = 1 and cos x = 0. Tan x has an asymptote (1 / 0) ■ At x = 180 degrees, sin x = 0 and cos x = 1. Tan x must be 0 (0 / 1) ■ At x = 270 degrees, sin x = 1 and cos x = 0. Tan x has an asymptote (1 / 0) Amplitude, Midline, and Period ○ The Amplitude is the height from the centerline to the peak. We can also measure the height from the highest to the lowest point and divide that by two. ○ The Vertical Shift is how far the function has shifted vertically from its original position. ○ The Period goes from one peak to the next peak (or from one point to the next matching point): ○ The Phase Shift is how far the function is shifted horizontally from its original position. ○ y = A sin(B(x + C)) + D ■ amplitude is A ■ period is 2π/B ■ phase shift is C ■ vertical shift is D Amplitude, Midline, and Period (continued) ○ Frequency: is how often something happens per unit of time (per "1"). ○ Period and Frequency are related: ■ Frequency = 1 / period ■ Period = 1 / frequency Amplitude, Midline, and Period (continued) ● Graphing sinusoidal functions ○ general period formula: ■ ω = ⟨regular period⟩ / ∣B∣ ○ For sines and cosines (and their reciprocals), the "regular" period is 2π, so their formula is: ■ period formula for sines & cosines: ● ω = 2π / ∣B∣ ○ For tangents and cotangents, the "regular" period is π, so their formula is: ■ period formula for tangents & cotangents: ■ ω = π / ∣B∣ Amplitude, Midline, and Period (continued) ○ Find the amplitude, period, phase shift, and vertical shift of s(t)=−2.5tan(4t−3π)−4 ■ The amplitude is given by the multiplier on the trig function. In this case, there's a –2.5 multiplied directly onto the tangent. ■ This is the "A" from the formula and means that the amplitude is 2.5. ■ The regular period for tangents is π. In this function, there's a 4 being multiplied by the variable, thus B = 4. ■ Plugging into the period formula, = π / 4 ■ To find the phase shift, isolate the variable with the shift value, now factor out the 4 (also known as "C") that's multiplied on the variable ● 4t − 3π = 4(t − 3π / 4) ■ the phase shift is ¾ π ● Sinusoidal models ○ Graphs of functions defined by y = sin x are called sine waves, or sinusoidal waves Trigonometric Equations and Identities Formulas to Know Coordinate Definitions for Trigonometric Functions