Question

Evaluate. $$\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$$

   Evaluate.
$$\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$$

Algebra and Trigonometry

Algebra and Trigonometry

Judith A. Beecher,… 4th Edition

Chapter 7, Problem 40

Instant Answer

Solved by Expert Ankit Gupta

07/08/2020

Step 1

This is because the inverse sine function, $\sin^{-1}(x)$, is defined only for the interval $[-1,1]$ and its range is $[-\frac{\pi}{2}, \frac{\pi}{2}]$. Show more…

Show all steps

lock

Thanks for your feedback!

Evaluate. $$\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$$

Play audio

Feedback

Powered by NumerAI

Ankit Gupta and 65 other subject Precalculus educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the basic trigonometric functions and are used to determine angles when given a trigonometric value. They undo the effect of the trigonometric functions, although care must be taken since many trigonometric functions are not one-to-one over their entire domains, necessitating restrictions for their inverses.

Principal Value and Range Restriction

The principal value of an inverse trigonometric function is the value within a specified range where the function is defined to be the inverse. For the inverse sine function, the principal value is defined such that its output falls within the interval [-?/2, ?/2]. This restriction ensures that each sine value corresponds to a unique angle, providing consistency across evaluations.

Trigonometric Function Symmetry

Trigonometric function symmetry, especially in the sine function, is a crucial concept. The sine function exhibits symmetry such as sin(?) = sin(? - ?), which is used to relate angles that may not initially lie within the principal value range of the inverse function to equivalent angles that do. This understanding helps in evaluating inverse functions when the original angle is outside the restricted interval.

Recommended Videos

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later