Question

(6.5) Evaluate arcsin $\left[\sin \left(\frac{5 \pi}{6}\right)\right]$.

   (6.5) Evaluate arcsin $\left[\sin \left(\frac{5 \pi}{6}\right)\right]$.

Precalculus

John W. Coburn 2nd Edition

Chapter 10, Problem 56

Instant Answer

Solved by Expert Charles Carter

05/25/2020

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Step 1: First, we need to understand that the arcsine function only returns values between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$. Show more…

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(6.5) Evaluate arcsin $\left[\sin \left(\frac{5 \pi}{6}\right)\right]$.

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Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, are functions that return the angle whose trigonometric value is the given number. They essentially 'undo' the operation of the corresponding trigonometric function by providing the principal value of the angle.

Principal Value and Range Restrictions

The inverse sine function (arcsin) has a principal value that is limited to a specific interval, typically [-?/2, ?/2]. This restriction is crucial because the original sine function is not one-to-one over its entire domain. When evaluating an expression like arcsin(sin(x)), the result must lie within this range, even if the original angle x does not.

Periodic Properties and Symmetry of Sine

The sine function is periodic with a period of 2?, and it exhibits symmetry properties that allow angles outside the principal value range to be 'mapped' back within it. This means that for an angle not in the interval [-?/2, ?/2], one can find an equivalent angle (with the same sine value) that does lie within the appropriate range for the arcsin function.

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