In Exercises 39–54, find the exact value of each expression, if p...

Question

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator.

tan⁻¹ (tan 2π/3)

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the following trigonometric expression without the help of a calculator. The expression we are given is the inverse tangent of the tangent of three pi divided by four. There are four answer choices. First recall that the inverse property when we are using tangent is the inverse tangent of the tangent of X would equal X as long as X is in the interval negative pi divided by two to pi divided by two. So in this case, when we are looking at the inverse tangent of the tangent of three pi divided by four, three pi divided by four is not in the interval between negative pi divided by two and pi divided by two. So what does that mean? Well, that means that this does not equal three pi divided by four to find out what it does mean we need to first go to a unit circle and find out what is the value of the tangent of three pi divided by four? And the tangent of that is negative one. So now we can substitute negative one in or the tangent of three pi divided by four in our original expression. So we have the inverse tangent of negative one and this will equal some angle measure. We don't know I'm gonna call it alpha. So we are asking ourselves the tangent of what angle equals negative one. Well, you can refer back to that unit circle to help. And we find that the tangent of negative pi divided by four with equal negative one. So that means that the value of the inverse tangent of the tangent of three pi divided by four equals negative pi divided by four. That is our answer and it is answer choice B have a nice day.

In Exercises 39–54, find the exact value of each expression, if possible. Do not use a calculator. tan⁻¹ (tan 2π/3)

Key Concepts

Inverse Trigonometric Functions

Tangent Function and Its Periodicity

Reference Angles

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