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

this problem, we're asked to evaluate the expression the cosine of the inverse sine of 1. Now, because this is a composite function, we know that we want to look at that inside function first, which is the inverse sine of 1. Now when working with inverse trig functions, we also know that we can think of this as, okay, the sine of what angle is equal to 1. Now again, because this is an inverse trig function, we also know that we are only looking at angles within a specific interval. So for the inverse sine, we only wanna look at angles between negative pi over 2 and pi over 2, so only on this half of my unit circle. I wanna find the angle for which the sine is equal to 1 in this interval, which is pi over 2. So the sine of pi over 2 is equal to 1. So that tells me that my answer to this inside function is pi over 2. Now that I have that information, I'm just left to find the cosine of that angle, pi over 2, which is 0. So that gives me my final answer. The cosine of the inverse sine of 1 is equal to 0. Thanks for watching, and let me know if you have questions.

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Evaluate Composite Trig Functions

Trigonometry

5. Inverse Trigonometric Functions and Basic Trigonometric Equations

Evaluate Composite Trig Functions

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Multiple Choice

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

$0$

$1$

$-1$

$\frac12$

Verified step by step guidance

Understand the problem: We need to evaluate $ \cos(\sin^{-1}(1)) $. This involves understanding the inverse trigonometric function $ \sin^{-1} $ and the cosine function.

Recall that $ \sin^{-1}(x) $ gives the angle whose sine is $ x $. Therefore, $ \sin^{-1}(1) $ is the angle whose sine is 1.

Recognize that the sine of $ \frac{\pi}{2} $ (or 90 degrees) is 1. Thus, $ \sin^{-1}(1) = \frac{\pi}{2} $.

Substitute $ \frac{\pi}{2} $ into the cosine function: $ \cos(\sin^{-1}(1)) = \cos(\frac{\pi}{2}) $.

Recall that $ \cos(\frac{\pi}{2}) = 0 $. Therefore, the expression evaluates to 0.

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Evaluate the expression.cos(sin−11)\cos\left(\sin^{-1}1\right)cos(sin−11)

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Evaluate Composite Trig Functions practice set

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