Question
Participation Activity \#8 This is similar to Try It \#2e in the OpenStax text. Evaluate \( \cos ^{-1}\left(-\frac{1}{2}\right) \) Enter the exact answer.
Ask by Collins Davison. in the United States
Feb 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
cos⁻¹(–1/2) = 2π/3
Solution
To evaluate cos⁻¹(–1/2), we need to determine the angle θ in the interval [0, π] such that cos θ = –1/2.
Recall that cos(π/3) = 1/2. Since cosine is negative in the second quadrant (where angles lie between π/2 and π), the corresponding angle with cosine –1/2 is π – (π/3), which simplifies to 2π/3.
Thus, cos⁻¹(–1/2) = 2π/3.
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Simplify this solution The Deep Dive
To evaluate \( \cos^{-1}\left(-\frac{1}{2}\right) \), we want to find the angle whose cosine is \(-\frac{1}{2}\). In the unit circle, the cosine function is negative in the second quadrant. The angle that corresponds to \(-\frac{1}{2}\) is \( \frac{2\pi}{3} \) radians (or \(120^\circ\)). Therefore, the exact answer is: \[ \frac{2\pi}{3} \]
