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Chapter 4: Problem 33

Evaluate the trigonometric function using its period as an aid. $$\cos \frac{7 \pi}{3}$$

Short Answer

Expert verified
The value of \(\cos(\frac{7\pi}{3})\) is \(\frac{1}{2}\).

Step by step solution

01

Rewrite the angle

As a first step, let's rewrite the given angle \(\frac{7\pi}{3}\) in terms of \(2\pi\). We can represent \(\frac{7\pi}{3} = 2\pi + \frac{\pi}{3}\). Dividing \(2\pi\) into \(3\pi/3 = \pi\), we have \(\frac{7\pi}{3} = 2\pi + \frac{\pi}{3}\) or \(\frac{7\pi}{3} = 2\pi + \frac{\pi}{3} = \frac{6\pi}{3} + \frac{\pi}{3} = \frac{7\pi}{3}\). Therefore, \(\cos(\frac{7\pi}{3}) = \cos(2\pi + \frac{\pi}{3})\).
02

Apply the Periodicity of Cosine

The cosine function has a period of 2蟺, which means that adding a multiple of \(2\pi\) to any angle does not affect the function's value. Therefore, we can write: \(\cos(2\pi + \frac{\pi}{3}) = \cos(\frac{\pi}{3})\).
03

Compute the Value

Finally, we need to compute the value of \(\cos(\frac{\pi}{3})\). This is a standard value that we should know from the unit circle. It's equal to 1/2. Therefore, \(\cos(\frac{7\pi}{3}) = \cos(\frac{\pi}{3}) = \frac{1}{2}\).

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