Skip to content

Physics Playground

Simulations How we teach Pricing

Language

Log in Start learning free

Loading…

Physics Playground

Built on open, peer-reviewed sources: OpenStax, The Feynman Lectures on Physics, HyperPhysics, NASA and NIST. See our sources.

Learn

Physics course
Math for Physics
Start with the math toolkit
Simulations
Glossary
Exercises
Past exams
Library
How we teach

Company

Pricing
Terms
Privacy
Contact

© 2026 Physics Playground. Made for the curious.Language

Exercises Trigonometry

Example 9 from Algebra and Trigonometry, 8.3 Inverse Trigonometric Functions

Finding the Cosine of the Inverse Sine of an Algebraic Expression

Find a simplified expression for $\cos (\sin^{- 1} (\frac{x}{3}))$ for $- 3 \leq x \leq 3.$

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.

Solution
We know there is an angle $θ$ such that $\sin θ = \frac{x}{3} .$

$\begin{array}{l} \sin^{2} θ + \cos^{2} θ = 1 & Use the Pythagorean Theorem . \\ {(\frac{x}{3})}^{2} + \cos^{2} θ = 1 & Solve for cosine . \\ \cos^{2} θ = 1 - \frac{x^{2}}{9} \\ \cos θ = \pm \sqrt{\frac{9 - x^{2}}{9}} = \pm \frac{\sqrt{9 - x^{2}}}{3} \end{array}$
Because we know that the inverse sine must give an angle on the interval $[- \frac{π}{2}, \frac{π}{2}],$ we can deduce that the cosine of that angle must be positive.

$\cos (\sin^{- 1} (\frac{x}{3})) = \frac{\sqrt{9 - x^{2}}}{3}$

Your turn

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

Find a simplified expression for $\sin (\tan^{- 1} (4 x))$ for $- \frac{1}{4} \leq x \leq \frac{1}{4} .$

How did it go?

PreviousEvaluating the Composition of a Sine with an Inverse Tangent Next Graphing the Equations of an Identity

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.