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Exercises Trigonometry

Example 6 from Algebra and Trigonometry, 8.3 Inverse Trigonometric Functions

Evaluating the Composition of an Inverse Sine with a Cosine

Evaluate $\sin^{- 1} (\cos (\frac{13 π}{6}))$

ⓐby direct evaluation.
ⓑ by the method described previously.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. ⓐ Here, we can directly evaluate the inside of the composition.
  $\begin{array}{l} \begin{array}{l} \cos (\frac{13 π}{6}) = \cos (\frac{π}{6} + 2 π) \\ = \cos (\frac{π}{6}) \\ = \frac{\sqrt{3}}{2} \end{array} \end{array}$
  Now, we can evaluate the inverse function as we did earlier.
  
  $\sin^{- 1} (\frac{\sqrt{3}}{2}) = \frac{π}{3}$
2. ⓑ We have $x = \frac{13 π}{6}, y = \frac{π}{6},$ and
  $\begin{array}{r} \sin^{- 1} (\cos (\frac{13 π}{6})) = \frac{π}{2} - \frac{π}{6} \\ = \frac{π}{3} \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Evaluate $\cos^{- 1} (\sin (- \frac{11 π}{4})) .$

How did it go?

PreviousUsing Inverse Trigonometric Functions Next Evaluating the Composition of a Sine with an Inverse Cosine

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 8.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.