QuestionJune 6, 2025

Find the following in radians without using a calculator. cos^-1(-(sqrt (2))/(2))=([?]pi )/([ ])

Find the following in radians without using a calculator. cos^-1(-(sqrt (2))/(2))=([?]pi )/([ ])
Find the following in radians without using a calculator. cos^-1(-(sqrt (2))/(2))=([?]pi )/([ ])

Solution
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Answer

\frac{3\pi}{4} Explanation 1. Identify the angle The value -\frac{\sqrt{2}}{2} corresponds to \cos(\theta) where \theta is in the second quadrant. 2. Determine reference angle Reference angle for \cos(\theta) = \frac{\sqrt{2}}{2} is \frac{\pi}{4}. 3. Calculate angle in second quadrant In the second quadrant, \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}.

Explanation

1. Identify the angle<br /> The value $-\frac{\sqrt{2}}{2}$ corresponds to $\cos(\theta)$ where $\theta$ is in the second quadrant.<br />2. Determine reference angle<br /> Reference angle for $\cos(\theta) = \frac{\sqrt{2}}{2}$ is $\frac{\pi}{4}$.<br />3. Calculate angle in second quadrant<br /> In the second quadrant, $\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$.
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