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Evaluate Composite Trig Functions quiz

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  • What is the first step when evaluating a composite trig function like sin(cos^{-1}(1/2))?
    Evaluate the innermost function first; in this case, find cos^{-1}(1/2) before applying the sine function.
  • How do you interpret cos^{-1}(1/2) in terms of the unit circle?
    It asks, 'The cosine of what angle gives 1/2?' which is pi/3 on the unit circle.
  • What is the value of sin(cos^{-1}(1/2))?
    It is sqrt(3)/2, because cos^{-1}(1/2) = pi/3 and sin(pi/3) = sqrt(3)/2.
  • Why can't you always cancel a trig function with its inverse, such as in cos(cos^{-1}(11pi/6))?
    Because the inverse trig function only returns values within a specific interval, so the answer may not be the original input.
  • What is the correct answer to cos(cos^{-1}(11pi/6))?
    The answer is pi/6, not 11pi/6, because cos^{-1} only returns angles between 0 and pi.
  • What is the interval for the output of the inverse tangent function, tan^{-1}(x)?
    The interval is from -pi/2 to pi/2.
  • How do you evaluate cos(tan^{-1}(0))?
    First, tan^{-1}(0) = 0, then cos(0) = 1.
  • What happens if you try to evaluate sin(sin^{-1}(2))?
    It is undefined because 2 is outside the allowable range of [-1, 1] for the sine function.
  • How can you evaluate sin(tan^{-1}(3/4)) if 3/4 is not on the unit circle?
    Draw a right triangle with opposite side 3 and adjacent side 4, find the hypotenuse (5), and compute sin(theta) = 3/5.
  • When should you use a right triangle instead of the unit circle for composite trig functions?
    Use a right triangle when the argument is not a common unit circle value, such as fractions like 3/4 or -5/13.
  • How do you determine the correct quadrant for your triangle when evaluating sin(cos^{-1}(-5/13))?
    Since cos^{-1} returns angles in [0, pi] and the cosine is negative, the angle is in quadrant II.
  • What are the side lengths of the triangle used to evaluate sin(cos^{-1}(-5/13))?
    Adjacent side is -5, hypotenuse is 13, and the opposite side (found using the Pythagorean theorem) is 12.
  • What is the value of sin(cos^{-1}(-5/13))?
    It is 12/13, since the opposite side is 12 and the hypotenuse is 13.
  • Why is the answer to sin(cos^{-1}(-5/13)) positive even though the argument is negative?
    Because in quadrant II, sine values are positive.
  • What is the value of cos^{-1}(sin(pi/3))?
    It is pi/6, because sin(pi/3) = sqrt(3)/2 and cos^{-1}(sqrt(3)/2) = pi/6.