Math / Trigonometry

Question16) $\sec \cos ^{-1}-\sqrt{2}$

Studdy Solution

STEP 1

1. The expression involves trigonometric functions and their inverses.
2. We need to evaluate the expression $\sec(\cos^{-1}(-\frac{\sqrt{2}}{2}))$ .
3. The inverse cosine function, $\cos^{-1}$ , returns an angle whose cosine is the given value.
4. The secant function, $\sec$ , is the reciprocal of the cosine function.

STEP 2

1. Determine the angle from the inverse cosine function.
2. Use the angle to find the secant value.

STEP 3

Determine the angle $\theta$ such that:
$\cos(\theta) = -\frac{\sqrt{2}}{2}$
The angle $\theta$ is in the range $[0, \pi]$ for $\cos^{-1}$ . The cosine of $\theta$ is $-\frac{\sqrt{2}}{2}$ , which corresponds to:
$\theta = \frac{3\pi}{4}$

STEP 4

Calculate $\sec(\theta)$ using the angle $\theta = \frac{3\pi}{4}$ .
Since $\sec(\theta) = \frac{1}{\cos(\theta)}$ , we have:
$\sec\left(\frac{3\pi}{4}\right) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}}$
Simplify the expression:
$-\frac{2}{\sqrt{2}} = -\sqrt{2}$
The value of the expression is:
$\boxed{-\sqrt{2}}$

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