In Exercises 29–51, find the exact value of each expression. Do n...

Question

In Exercises 29–51, find the exact value of each expression. Do not use a calculator.

tan[sin⁻¹ (− 1/2)]

Accepted Answer

Hello, everyone. We are going to determine the exact value of the expression, the tangent of the arc cosine of radical two divided by two. Without using a calculator, we have four answer choices. We will be using a unit circle quite frequently. So if you need to find yours, please do. So. Now first, I'm gonna substitute the arc cosine of radical two divided by two for the angle theta. This means that I can find the value of the A cosine of radical two divided by two and then we'll plug it back in. So taking the cosine of both sides, I get that the cosine of theta must equal radical two divided by two, referencing my unit circle. My options for the are pi divided by four, the end seven pi divided by four. However, I know the range for when Y equals the arc cosine of X. This has a range of zero to pie including zero and pie and seven pi divided by four does not fit in that range. So this must be theta equaling pi divided by four. So now if we know that the a cosine of radical two divided by two was theta and that theta is pi divided by four. We can substitute that in and find the tangent of pi divided by four to do. So using our unit circle, we must recall that tangent is the same as the sign of in this case, pi divided by four divided by the cosine of the angle pi divided by four. So now going to my unit circle, the sign of pi divided by four is radical two divided by two. The cosine of pi divided by four is radical two divided by two. Since the numerator and denominator of this fraction are the same, this reduces to one. So in the end, the tangent of the a cosine of radical two divided by two is one. So our value is one, our answer choice is a have a nice day.

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan[sin⁻¹ (− 1/2)]

Key Concepts

Inverse Trigonometric Functions

Trigonometric Ratios

Unit Circle

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