Evaluate the expression. cos ( sin − 1 ( − 7 25 ) ) \cos\left(\sin^{-1}\left(-\frac{7}{25}\right)\right) cos ( sin − 1 ( − 25 7 ) )

this problem, we're asked to evaluate the expression the cosine of the inverse sine of negative 7 over 25. Now jumping right into our steps here, we're going to take our inside function and use its interval in order to identify what quadrant our triangle will go in. Now here, my inside function is the inverse sine. And when dealing with the inverse sine, I know that I can only work with my angle interval between negative pi over 2 and positive pi over 2. So that is either quadrant 4 or quadrant 1. Now here, considering the sign of our argument, here we're taking the inverse sign of a negative value. Now within this interval from negative pi over 2 to positive pi over 2, my sign values can only be negative down here in quadrant 4. So that's where my right triangle will go. Now step 1 is done. We can move on to step 2 and actually draw that right triangle in that quadrant 4. So I'm gonna go ahead and draw my triangle here. Once I have drawn my triangle, we can use our argument in order to label our angle theta and two sides. Now, here, the orientation of my triangle is different than it would be in quadrants 1, 2, or 3 because it's in quadrant 4. Remember, we're always drawing our triangles from our x x axis so that that's where our angle measures from. Now here's my angle theta, and I can go ahead and use that argument negative 7 over 25, which tells me that the sine of that angle I just labeled is equal to negative 7 over 25. Now using these values and SOHCAHTOA, this tells me that my opposite side is negative 7, and my hypotenuse is 25. Now here, when we look at this, since we have a negative value, we kind of want to double check this with our location. And having this side length as negative makes sense because we are in quadrant 4. In quadrant 4, my y values are negative. So it makes sense that this side length is negative 7. Now step 2 is done. Moving on to step 3, we're going to use the Pythagorean theorem to find that missing third side. Now our third side that's missing here is a leg length, so we're either solving for a or b. It doesn't matter which one. So setting up my pythagorean theorem, a squared plus b squared equals c squared. I'm gonna be solving for a squared here. So adding in that negative 7 squared, that's equal to c squared to 25 squared. So simplifying this, a squared plus 49 gives me 25 squared and squaring that 25 gives me a value of 625. Subtracting 49 from both sides here cancels that 49. It leaves me with a squared is equal to 625 minus 49, which is 576. Some pretty large numbers happening here. But our last step is to take the square root of both sides, and that will give me a value of 24. Now I can go ahead and label that missing side length on my triangle. 24, it makes sense that this value is positive. Here in quadrant 4, my x values are positive. Now step 3 is done. We can move on to step number 4, which is to actually use our triangle to evaluate that outside function. Now here, my outside function is the cosine. And in order to find the cosine of my angle theta, I need to take my adjacent side and divide it by my hypotenuse based on SOHCAHTOA. Now here my adjacent side is 24, my hypotenuse is 25. So 24 over 25 is my final answer. The cosine of the inverse sine of negative 7 over 25 is positive 24 over 25. Now, again, we can always double double check what's going on with our signs here. Here in quadrant 4, my cosine values are positive based on my mnemonic device. All students take calculus here in quadrant 4. My cosine values are positive. It makes sense that I ended up with a positive answer. So that's all for this one. Thanks for watching, and I'll see you in the next one.

Evaluate the expression. cos ⁡ ( sin ⁡ − 1 ( − 7 25 ) ) \cos\left(\sin^{-1}\left(-\frac{7}{25}\right)\right) cos ( sin − 1 ( − 25 7 ) )

− 24 25 -\frac{24}{25} − 25 24

11. Inverse Trigonometric Functions and Basic Trig Equations

Evaluate Composite Trig Functions

Precalculus

11. Inverse Trigonometric Functions and Basic Trig Equations

Evaluate Composite Trig Functions

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Multiple Choice

Evaluate the expression.
$\cos\left(\sin^{-1}\left(-\frac{7}{25}\right)\right)$

$\frac{7}{24}$

$\frac{8}{25}$

$\frac{24}{25}$

$-\frac{24}{25}$

Verified step by step guidance

Recognize that the expression involves the composition of inverse trigonometric functions. Specifically, we have $ \cos(\sin^{-1}(-\frac{7}{25})) $.

Understand that $ \sin^{-1}(x) $ gives an angle $ \theta $ such that $ \sin(\theta) = x $. Here, $ \sin(\theta) = -\frac{7}{25} $.

Use the Pythagorean identity $ \sin^2(\theta) + \cos^2(\theta) = 1 $ to find $ \cos(\theta) $. Substitute $ \sin(\theta) = -\frac{7}{25} $ into the identity.

Calculate $ \cos^2(\theta) = 1 - \left(-\frac{7}{25}\right)^2 $. Simplify this expression to find $ \cos^2(\theta) $.

Take the square root to find $ \cos(\theta) $. Since $ \theta $ is in the range of $ \sin^{-1} $, which is $[-\frac{\pi}{2}, \frac{\pi}{2}]$, $ \cos(\theta) $ will be positive. Thus, $ \cos(\theta) = \frac{24}{25} $.