Evaluate the expression. cos − 1 ( cos ( π 2 ) ) \cos^{-1}\left(\cos\left(\frac{\pi}{2}\right)\right) cos − 1 ( cos ( 2 π ) )

Here we're asked to evaluate the expression, the inverse cosine of the cosine of pi over 2. Now right off the bat, you may be tempted to just go ahead and cancel that inverse cosine and that cosine. But remember, we have to be extra careful here. So let's go ahead and solve this the way we would any composite trig function starting with that inside function, the cosine of pi over 2. Now coming over to our unit circle, looking at my angle pi over 2, I know that my cosine is equal to 0. So my inside function gives me 0, and I'm just left to find the inverse cosine of that 0. Now remember when considering inverse trig functions, we can also think of this as, okay, the cosine of what angle is going to give me 0. And specifically, that angle has to be within my specified interval for the inverse cosine, which happens to be from 0 to pi. So looking at my unit circle over here, I'm looking for an angle between 0 and pi for which the cosine is equal to 0. Now the angle for which that happens to be true is pi over 2. That is within my specified interval from 0 to pi, and the cosine is 0. So here, specified interval from 0 to pi, and the cosine is 0. So here, my answer is pi over 2. Now here, you may have noticed right away that pi over 2 is within your specified interval for the inverse cosine. So if you notice that from the beginning, remember that you actually can cancel the inverse cosine and the cosine to get your final answer here. But remember to be careful and always, always double check that interval. Now here, the inverse cosine of the cosine of pitwo is simply pitwo. Thanks for watching. And let me know if you have any questions.

0. Functions

Inverse Trigonometric Functions

Calculus

0. Functions

Inverse Trigonometric Functions

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Multiple Choice

Evaluate the expression.
$\cos^{-1}\left(\cos\left(\frac{\pi}{2}\right)\right)$

$0$

$\frac{\pi}{2}$

$\pi$

Undefined

Verified step by step guidance

Understand the problem: We need to evaluate the expression $ \cos^{-1}(\cos(\frac{\pi}{2})) $. This involves understanding the properties of the inverse cosine function.

Recall the definition of the inverse cosine function: $ \cos^{-1}(x) $ is the angle $ \theta $ such that $ 0 \leq \theta \leq \pi $ and $ \cos(\theta) = x $.

Evaluate the inner function: Calculate $ \cos(\frac{\pi}{2}) $. Since $ \frac{\pi}{2} $ is 90 degrees, $ \cos(\frac{\pi}{2}) = 0 $.

Substitute the result into the inverse function: We now have $ \cos^{-1}(0) $.

Determine the angle: Find the angle $ \theta $ such that $ \cos(\theta) = 0 $ and $ 0 \leq \theta \leq \pi $. The angle that satisfies this condition is $ \frac{\pi}{2} $.