In Exercises 63–82, use a sketch to find the exact value of each ...

Question

In Exercises 63–82, use a sketch to find the exact value of each expression.

cos (sin⁻¹ 4/5)

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the following trigonometric expression by sketching the expression is the cosine of the inverse sign of 35 37th. We have four possible answer choices. First, I'm drawing myself the X and Y axis that I'm going to use for my sketch. Next, I'm gonna say that the angle alpha is gonna be equal to the inverse sign of 37th. This way I can find the value of alpha and work with it when I need to use the cosign. So the inverse sign can be undone by the sign. So this will work out to be the sign of alpha equals 35 37th. Since sin is positive, we know this is gonna be in the first quadrant. And I recall that this is the ratio of Y to R. So the side Y versus the side R which will be our hypotenuse or a radius if we were working in a circle. So I wanna draw this onto my sketch. So why means I'm going up about and R means that is my diagonal and that will be 37 an alpha will be in here as an angle right now. I don't know the value for X. I can find it using the Pythagorean theorem. So X squared plus Y squared equals R squared. So X squared plus 35 squared equals 37 squared. So X squared plus equals 1369. Subtracting 1225 from both sides. X squared will equal taking the square root of both sides X would equal 12. Since we are working in quadrant one, X is positive, so we can keep this as a positive 12. So now we want to find the cosine of alpha and that is gonna be the value of X divided by the value of R. So this will be 12 divided by 37. And that's our answer. The exact value here is 12 37th which is answer choice B have a nice day.

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In Exercises 63–82, use a sketch to find the exact value of each expression. cos (sin⁻¹ 4/5)