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Chapter 0: Problem 42

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

Short Answer

Expert verified
The exact value of the given expression is \(135^\circ\) (or \(\frac{3\pi}{4}\) radians).

Step by step solution

01

Recognize the value

We are given with the expression: \(\cos^{-1}\left(-\frac{1}{\sqrt{2}}\right)\). It means that we are trying to find an angle whose cosine value is \(-\frac{1}{\sqrt{2}}\), and remember that the angle solution will lie in either the second or third quadrant, as cosine is negative there.
02

Find the angle with the given cosine value

Recall the values of cosine for common angles. Cosine is negative in the second and third quadrants, so look for the angle \(\theta\) in the second quadrant where the cosine value is positive. In the first quadrant, the angle \(45^\circ\) (or \(\frac{\pi}{4}\) in radians) has a cosine value of \(\frac{1}{\sqrt{2}}\). Since in the second quadrant, the cosine is negative, the angle with the cosine value -\(\frac{1}{\sqrt{2}}\) will be the supplement of \(45^\circ\).
03

Find the supplement of the angle

To find the supplement of \(45^\circ\) (or \(\frac{\pi}{4}\) in radians), subtract it from \(180^\circ\ (or \pi\, radians). \[ 180^\circ - 45^\circ = 135^\circ\\ \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]
04

State the answer

We found the angle \(\theta\) whose cosine value is \(-\frac{1}{\sqrt{2}}\). The solution to the given expression is: \[ \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right) = 135^\circ \,(\text{or} \,\frac{3\pi}{4} \, \text{radians}) \]

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Most popular questions from this chapter

Let $$ f(x)=\left\\{\begin{array}{ll} 2 x-1 & \text { if } x<1 \\ \sqrt{x} & \text { if } 1 \leq x<4 \\ \frac{1}{2} x^{2}-6 & \text { if } x \geq 4 \end{array}\right. $$ Find \(f^{-1}(x)\), and state its domain.

Find the exact value of the given expression. $$ \csc ^{-1} \sqrt{2} $$

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=\frac{5 x}{x-1}+5 x $$

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You are given the graph of a function \(f .\) Determine whether \(f\) is one-to- one.
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