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Chapter 0: Problem 36

Find the exact value of the given expression. $$ \cos ^{-1} 0 $$

Short Answer

Expert verified
The exact value of the given expression is \(\cos^{-1}(0) = \frac{蟺}{2}\).

Step by step solution

01

Recall the definition of the inverse cosine function

The inverse cosine function, \(\cos ^{-1}\), is defined as the angle 胃 for which the cosine of 胃 is equal to a given value. In other words, if \(\cos(\theta) = x\), then \(\cos^{-1}(x) = \theta\). We are asked to find the angle \(\theta\) when the cosine of 胃 is 0, that is, we want to find \(\cos^{-1}(0)\).
02

Analyze the unit circle

We can use the unit circle to find the angle 胃 for which the cosine is equal to 0. The cosine function on the unit circle corresponds to the x-coordinate of a point on the circle. We want to find an angle 胃 such that the x-coordinate on the unit circle is 0. This occurs on the circle at two points: (0,1) and (0,-1). The coordinates correspond to the angles \(胃 =\frac{蟺}{2}\) and \(胃 =\frac{3蟺}{2}\), but since the inverse cosine function returns angles in the range of \(0 \leq 胃 \leq 蟺\), we choose the angle \(胃 =\frac{蟺}{2}\).
03

Write the final answer

Therefore, the exact value of the given expression is \[ \cos ^{-1} (0) = \frac{蟺}{2} \]

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Most popular questions from this chapter

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=\sin x, \quad y=2 \sin \frac{x}{2}\)

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Plot the graph of the function \(f\) in (a) the standard viewing window and (b) the indicated window. $$ f(x)=x^{4}-2 x^{2}+8 ; \quad[-2,2] \times[6,10] $$

Show that \(f\) and \(g\) are inverses of each other by verifying that \(f[g(x)]=x\) and \(g[f(x)]=x\). $$ f(x)=\frac{1+x}{1-x} ; \quad g(x)=\frac{x-1}{x+1} $$

Find the zero(s) of the function f to five decimal places. $$ f(x)=\sin 2 x-x^{2}+1 $$
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