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Chapter 4: Problem 17

Evaluate the expression without using a calculator. $$ \arcsin \frac{\sqrt{2}}{2} $$

Short Answer

Expert verified
The evaluated expression is \( \frac{\pi}{4} \) or \( 45^\circ \).

Step by step solution

01

Identify Corresponding Angle

Remember that for arcsine, we're looking for the angle whose sine is \( \frac{\sqrt{2}}{2} \). From the knowledge of standard angles and unit circle, we know that the value of sine is \( \frac{\sqrt{2}}{2} \) when the angle is \( \frac{\pi}{4} \) or \( 45^\circ \)
02

Return the Angle

Since the range of the arcsine function is from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), the corresponding angle can only be \( \frac{\pi}{4} \), our final answer.

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Most popular questions from this chapter

Using calculus, it can be shown that the secant function can be approximated by the polynomial $$\sec x \approx 1+\frac{x^{2}}{2 !}+\frac{5 x^{4}}{4 !}$$ where \(x\) is in radians. Use a graphing utility to graph the secant function and its polynomial approximation in the same viewing window. How do the graphs compare?

Evaluate the expression without using a calculator. $$ \arccos \left(-\frac{1}{2}\right) $$

Use a graphing utility to graph the function. Include two full periods. $$ y=0.1 \tan \left(\frac{\pi x}{4}+\frac{\pi}{4}\right) $$

Evaluate the expression without using a calculator. $$ \arcsin \frac{1}{2} $$

Use a calculator to evaluate the expression. Round your result to two decimal places. $$ \arcsin 0.65 $$
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