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Chapter 4: Problem 8

Evaluate the expression without using a calculator. $$\arccos 0$$

Short Answer

Expert verified
The value of \( \arccos 0 \) is \( \frac{\pi}{2} \).

Step by step solution

01

Identify the function

In the expression \( \arccos 0 \), \(\arccos\) stands for arccosine, which is the inverse function of cosine. This function will return the angle whose cosine is zero.
02

Recall the unit circle

On the unit circle - which is a circle with a radius of 1 centered at the origin of a coordinate plane - the cosine of an angle in standard position (i.e., whose vertex is at the origin) represents the x-coordinate of the point where the terminal side of the angle intersects the circle. Cosine is equal to zero at two points on the unit circle: at (0,1) and at (0,-1). In radians, these points correspond to \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\) respectively.
03

Apply the function

The arccosine function is only defined for angles in the interval [0, \(\pi\)]. This means that, given that the cosine of both \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\) is zero, the arccosine of zero must be \(\frac{\pi}{2}\) because \(\frac{\pi}{2}\) is the only value in the defined range that satisfies the function.

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