Question
What is the exact value of cos^-1(- square root 2 over 2
Answers
Answered by
GPT-4o mini
To find the exact value of \(\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right)\), we need to determine the angle \( \theta \) such that:
\[
\cos(\theta) = -\frac{\sqrt{2}}{2}
\]
The cosine function is negative in the second and third quadrants. Specifically, the angles where the cosine of \( \theta \) equals \(-\frac{\sqrt{2}}{2}\) are:
\[
\theta = \frac{3\pi}{4} \quad \text{(in the second quadrant)}
\]
\[
\theta = \frac{5\pi}{4} \quad \text{(in the third quadrant)}
\]
However, the range of the \(\cos^{-1}(x)\) function is limited to \([0, \pi]\), which means we only consider the angle in the second quadrant:
\[
\theta = \frac{3\pi}{4}
\]
Thus, the exact value of \(\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right)\) is:
\[
\frac{3\pi}{4}
\]
\[
\cos(\theta) = -\frac{\sqrt{2}}{2}
\]
The cosine function is negative in the second and third quadrants. Specifically, the angles where the cosine of \( \theta \) equals \(-\frac{\sqrt{2}}{2}\) are:
\[
\theta = \frac{3\pi}{4} \quad \text{(in the second quadrant)}
\]
\[
\theta = \frac{5\pi}{4} \quad \text{(in the third quadrant)}
\]
However, the range of the \(\cos^{-1}(x)\) function is limited to \([0, \pi]\), which means we only consider the angle in the second quadrant:
\[
\theta = \frac{3\pi}{4}
\]
Thus, the exact value of \(\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right)\) is:
\[
\frac{3\pi}{4}
\]