Question

Evaluate. $$\sin ^{-1}\left(\sin \frac{7 \pi}{6}\right)$$

   Evaluate.
$$\sin ^{-1}\left(\sin \frac{7 \pi}{6}\right)$$

Algebra and Trigonometry

Algebra and Trigonometry

Judith A. Beecher,… 4th Edition

Chapter 7, Problem 53

Instant Answer

Solved by Expert Ankit Gupta

07/08/2020

Step 1

We know that $\sin \frac{7 \pi}{6}$ is equal to $-\sin \frac{\pi}{6}$ because $\sin(\pi - \theta) = \sin \theta$ and $\frac{7 \pi}{6}$ is $\pi + \frac{\pi}{6}$. Show more…

Show all steps

lock

Thanks for your feedback!

Evaluate. $$\sin ^{-1}\left(\sin \frac{7 \pi}{6}\right)$$

Play audio

Feedback

Powered by NumerAI

Ankit Gupta and 94 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, such as the arcsin, are designed to undo the original trigonometric function. They retrieve an angle whose sine, cosine, etc., equals a given value. However, because trigonometric functions are periodic and not one-to-one over their entire domain, their inverses are defined with restricted ranges to make them proper functions.

Principal Value of Arcsin

The arcsin function is defined to return values only within its principal value range of [-?/2, ?/2]. This restriction ensures that each value of sine corresponds uniquely to an angle, and it is critical when evaluating expressions like sin?¹(sin?), especially when ? lies outside this interval.

Periodicity of the Sine Function

The sine function is periodic with a period of 2?, meaning that sin? is the same as sin(? + 2?k) for any integer k. This property is important when evaluating inverse trigonometric functions because it helps identify an equivalent angle within the principal value range that has the same sine as the given angle.

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