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Chapter 4: Problem 102

In Exercises \(99-104,\) find two values of \(\theta, 0 \leq \theta<2 \pi,\) that satisfy each equation. $$\cos \theta=-\frac{1}{2}$$

Short Answer

Expert verified
The two possible values of \( \theta \) that satisfy the equation \( \cos \theta = -\frac{1}{2} \) in the range \( 0 \leq \theta < 2 \pi \) are \( \theta_1 = \frac{2 \pi}{3} \) and \( \theta_2 = \frac{4 \pi}{3} \).

Step by step solution

01

Finding the reference angle

The reference angle is found by taking the cosine inverse of the absolute value of \( \frac{1}{2} \), i.e. \( \theta_0 = \cos ^{-1} \left(\frac{1}{2}\right) \). This gives \( \theta_0 = \frac{\pi}{3} \) or \(60^\circ\).
02

Finding angles in the 2nd quadrant

In the second quadrant, the angle \( \theta \) is given by \( \pi - \theta_0 \). So, \( \pi - \frac{\pi}{3} = \frac{2 \pi}{3} \). Thus, one solution is \( \theta_1 = \frac{2 \pi}{3} \).
03

Finding angles in the 3rd quadrant

In the third quadrant, the angle \( \theta \) is given by \( \pi + \theta_0 \). So, \( \pi + \frac{\pi}{3} = \frac{4 \pi}{3} \). Thus, the second solution is \( \theta_2 = \frac{4 \pi}{3} \).

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