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Chapter 0: Problem 47

Find the exact value of the given expression. $$ \sin \left(\sin ^{-1} \frac{1}{\sqrt{2}}\right) $$

Short Answer

Expert verified
The exact value of the given expression is \(\frac{\sqrt{2}}{2}\).

Step by step solution

01

Recall the definition of inverse trigonometric functions

Inverse trigonometric functions are defined such that when the original trigonometric function is applied to its inverse, they cancel each other out, and we are left with the input value of the inverse function. In this case, we have the sine function and its inverse: \( \sin (\sin^{-1}(x)) = x \)
02

Apply the definition to the given expression

Now, we can apply the definition to the given expression: \( \sin \left(\sin ^{-1} \frac{1}{\sqrt{2}}\right) = \frac{1}{\sqrt{2}} \)
03

Simplify the expression

The exact value of the expression is already in simplified form. However, we can also rationalize the denominator by multiplying the numerator and the denominator by \(\sqrt{2}\): \[ \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \] So, the exact value of the given expression is \(\frac{\sqrt{2}}{2}\).

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