Evaluate the expression. cos − 1 ( − 1 ) \cos^{-1}\left(-1\right) cos − 1 ( − 1 )

In this problem, we're asked to evaluate the expression the inverse cosine of negative one. Now remember you can think of this as okay the cosine of what angle is equal to negative one and we're looking for the angle for which that is true. Now coming over to our unit circle here, I know that we're taking the inverse cosine of a negative value. So within my interval from 0 to pi, I know that I'm looking for an angle specifically in this second quadrant where my cosine values are negative. Now in this second quadrant, looking at my quadrantal angle of pi, I know that the cosine of pi is equal to negative one. So this represents my solution. Theta is equal to pi if the inverse cosine is equal to negative one. Thanks for watching and I'll see you in the next one.

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse Sine, Cosine, & Tangent

Trigonometry

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Inverse Sine, Cosine, & Tangent

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Multiple Choice

Evaluate the expression.
$\cos^{-1}\left(-1\right)$

$0$

$\pi$

$\frac{\pi}{2}$

$\frac{3\pi}{2}$

Verified step by step guidance

Understand that $ \cos^{-1}(x) $ is the inverse cosine function, which gives the angle whose cosine is $ x $.

Recall that the range of the inverse cosine function, $ \cos^{-1}(x) $, is $[0, \pi] $. This means the output will be an angle in radians between 0 and $ \pi $.

Identify that the expression $ \cos^{-1}(-1) $ asks for the angle whose cosine is $-1$.

Recall that the cosine of $ \pi $ is $-1$. Therefore, $ \cos(\pi) = -1 $.

Conclude that the angle whose cosine is $-1$ within the range $[0, \pi] $ is $ \pi $.