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Chapter 5: Problem 57

Write the trigonometric expression as an algebraic expression. $$\sin (\arcsin x+\arccos x)$$

Short Answer

Expert verified
The trigonometric expression \(\sin (\arcsin x + \arccos x)\) simplifies to 1 as an algebraic expression.

Step by step solution

01

Understanding the identity

Before simplifying, it's important to understand an identity regarding inverse sine and cosine functions, which is \(\arcsin x + \arccos x = \frac{\pi}{2}\) regardless of the value of x. This identity states that the sum of the arcsine and arccosine of a same variable x is always equal to \(\frac{\pi}{2}\).
02

Applying the identity

Given the trigonometric expression \(\sin (\arcsin x + \arccos x)\), we can directly replace the \(\arcsin x + \arccos x\) with \(\frac{\pi}{2}\). By doing so, we get \(\sin \frac{\pi}{2}\) as new expression.
03

Simplifying the expression

Lastly, the sine of \(\frac{\pi}{2}\) radians (or 90 degrees) is a well known value and is equal to 1. Therefore, the final algebraic expression is 1.

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