Trigonometric Functions #5 - Questions and Answers

Question 1 Find the exact value of sin 45 degrees, please. Answer 1 sin 45° = 1/√2 And we can rationalize the fraction: = √2 / 2 Answer 2 sin of 45 degrees is √2 / 2 Answer 3 sin 45° = √2 / 2 Question 2 The trigonometric functions sec (225°) Answer To find the value of the trigonometric functions, first, convert sec function in terms of cos (θ). Use the identity sec (θ) = 1 / cos (θ) We have θ = 225° = 180° + 45° cos (225°) = cos (180° + 45°) {cos (180 + θ) = -cos (θ)} = -cos (45°) = - 1 / √2 Now sec (θ) = 1 / cos (θ) sec (225°) = 1 / cos (225°) = 1 / ( -1 / √2 ) = -√2 = -1.4142 Question 3 Use the figures to find the exact value of the trigonometric function tan 2θ. Answer Using the triangle, tan θ = 3/4 Apply the formula of the trigonometric function, tan 2θ = (2tan θ) / (1 – tan²θ) Substitute the value of tan θ = 3/4 in the formula, tan 2θ = 2*(3/4) / (1 – (3/4)²) Solving the equation above, tan 2θ = 3/2 / 1 – 9/16 = 3/2 / 7/16 = 3/2 * 16/7 = 24/7 Thus the value of the trigonometric function tan 2θ is 24/7. Question 4 The two trigonometric functions defined for all real numbers are the ? function and the ? function. The domain of each of these functions is?. Answer 1 The sine function is one of the three primary functions in trigonometry, nthe others being cosine and tan functions. The sine x or sine theta can be defined as the ratio of the opposite side of a right triangle to its hypotenuse. and Cos function (or cosine function) in a triangle is the ratio of the adjacent side to that of the hypotenuse. The cosine function is one of the three main primary trigonometric functions and it is itself the complement of sine (cos + sine). The domain of a function is the set of all possible inputs for the function. For example: the domain of f(x) = x² is all real numbers, and the domain of g(x) = 1/x is all real numbers except for x = 0. The two trigonometric functions defined for all real numbers are the sine function and the cosine function. The domain of each of these functions is (-∞, ∞). Answer 2 Explanation: The sine function (sin) is defined as the ratio of the length of the side opposite a given angle in a right triangle to the length of the hypotenuse. In a unit circle (a circle with a radius of 1 unit), the sine of an angle is equal to the y-coordinate of the point on the unit circle that corresponds to that angle. The cosine function (cos) is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle. In a unit circle, the cosine of an angle is equal to the x-coordinate of the corresponding point on the unit circle. The domain of both the sine and cosine functions is the set of all real numbers. This means that you can plug in any real number as an angle into these functions, and they will output a real number as a result. The values returned by the sine and cosine functions can be positive or negative, depending on the quadrant in which the angle lies. Question 5 When using the half-angle formulas for trigonometric functions of α/2 I determine the sign based on the quadrant in which α lies.Determine whether the statement makes sense or does not make sense, and explain your reasoning. Answer For example: The trigonometric function is given as, cos (112.5°) cos (112.5°) = cos(225°/2) Here, α = 225° Apply the half angle formula for the given trigonometric function, cos(112.5°) in which the angle 112.5° lies in the quadrant (II) where sine and cosecant is positive. We have to determine the value of cos (112.5°) which is negative in second quadrant. Therefore in half angle formula, negative sign has to be put. cos(α/2) = -√((1+cosα)/2) The value of cos (112.5°) = cos(225°/2) cos (225°/2) = -√((1+cos 225°)/2) cos (225°/2) = -√((1-√2/2)/2) cos (225°/2) = -√(2-√2)/2)