In Exercises 14–19, use a sum or difference formula to find the e...

Question

In Exercises 14–19, use a sum or difference formula to find the exact value of each expression.

5𝝅
tan ---------
12

Accepted Answer

Hello, everyone. We are asked to use a sum or different formula and determine the exact value of the given trigonometric expression. The expression we are given is the tangent of seven pi divided by 12. There are four answer choices. First to decide if we are using a sum or a difference formula. We need to decide what numbers either add to seven pi divided by 12 or their difference is seven pi divided by 12. I think addition is gonna be the easiest way to go here and pi divided by three plus pi divided by four does add up to seven pi divided by 12. So I'm gonna use a sum formula because now I'm gonna be working with the concept that the tangent of pi divided by three plus pi divided by four is what I'm looking for. The value of. We're calling our sum formula for tangent. It states that the tangent of A plus B equals the tangent of A plus the tangent of B divided by one minus the tangent of A multiplied by the tangent of B. So in this example, A is gonna be pi divided by three B is gonna be pi divided by four. So I'm gonna fill that in to my formula. So the tangent of pi divided by three plus pi divided by four equals the tangent of pi divided by three plus the tangent of pi divided by four, all of it divided by one minus the tangent of pi divided by three, multiplied by the tangent of pi divided by four. So next I'm gonna go reference um either a unit circle or whatever I need to, to find the exact value of the tangent of pi divided by three. And when I do so I know that the exact value of the tangent of pi divided by three is radical three and the tangent of pi divided by four is one. So now I'm going to take those exact values and plug it in to our formula. So the tangent of Pi divided by three. So radical three plus the tangent of pi divided by four, which is one divided by one minus the tangent of pi divided by three, which is radical three multiplied by the tangent of pi divided by four, which is one. So we now have radical three plus one divided by one minus radical three. I wanna rationalize my denominator and I can do that by multiplying top and bottom by the conjugate which again has the same terms as the denominator but with the opposite sign in between them. So we are multiplying top and bottom by one plus radical three. And when we do so our numerator, this is kind of like one plus radical three squared as we can rearrange our addition. So that would give me scoring the first term one scoring. The last term will give me three, taking the product of the two terms and double, it would be plus two radical three divided by my denominator. The purpose with the conjugates is to end up with the difference of squares and which will cancel out our radical. So scoring the first term one squared is one minus the last term squared. So the square root of three squared which is three, I'm going to clean up my like terms. So in the numerator, I have four plus two radical three in the denominator, I have negative two, I'm gonna reduce my fraction. I can divide four by negative two and get negative two and I can divide two radical three by negative two and get minus radical three. So I'm going to rewrite this in the opposite order because it looks like all of our answer choices have the radical first. So this would be negative radical three minus two. This is the exact value of the tangent of seven pi divided by 12 and it matches answer choice. A have a nice day.

In Exercises 14–19, use a sum or difference formula to find the exact value of each expression. 5𝝅 tan --------- 12

Key Concepts

Sum and Difference Formulas

Tangent Function

Exact Values of Trigonometric Functions

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