In Exercises 9–16, use the given triangles to evaluate each expre...

Question

In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

Two right triangles labeled with angles and side lengths for evaluating trigonometric functions.

tan 𝜋/4 + csc 𝜋/6

Accepted Answer

Hello, everyone. We are asked to use the given triangles and find the value of the expression, rationalizing the denominator of the final answer. If necessary, we have two right A right triangles, we have a smaller right triangle called ABC with engle A as 45 degrees angle C is 45 degrees angle B is our right angle sides A B and BC both have a value of one in the hypotenuse. AC has a value of radical two. We have a slightly larger right triangle. Also named ABC where angle A is 30 degrees angle C is 60 degrees angle B is our right angle side. A B is radical three BC has a value of one and AC or hypotenuse has a value of two. In this problem, we are looking to find the tangent of pi divided by six plus the of pi divided by four. We have four answer choices first. Before I do anything else, I want to convert these radiant measures into degrees. So I know I could do that by substituting degrees in for pie. So this could be the tangent of 180 degrees divided by six plus the Coast Sea can of degrees divided by four. And this will be the tangent of 30 degrees plus the Coast Sea can of 45 degrees. This is a little easier for us to be able to locate on these triangles. Next, I'm gonna find the value of both the tangent of 30 degrees and the coy can of 45 degrees separately before I add them together. So starting with the tangent of degrees, I recall that tangent is the leg opposite the angle divided by the leg adjacent to the angle, 30 degrees is in our larger triangle as angle A opposite angle A, we have side BC which has a value of one and adjacent. We have side A B which has a value of radical three. So one divided by radical three, I'm going to rationalize that by multiplying top and bottom by radical three. So the tangent of 30 degrees ends up being radical three divided by three. Next, the Coast Sea can of 45 degrees. I recall that Coy can is the reciprocal of sign. I know that sine is the opposite divided by the hypotenuse. So this will be the hypotenuse divided by the opposite side. 45 degrees is in our smaller triangle. It is both angles A and C. So we can use either I'm gonna use angle A, the hypotenuse is radical too, the side opposite angle A is one. So this would be radical too. So now that I know the values of both pieces, I'm gonna put them back into the original and see what we get. So we have tangent of 30 degrees is three or sorry. It's radical three divided by three plus the coy can of 45 degrees is radical too. And there's no other way to combine these. So this is our answer. The tangent of pi divided by six plus the of pi divided by four equals radical three, divided by three plus radical two. And that is answer choice C have a nice day.

In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.
tan 𝜋/4 + csc 𝜋/6

Key Concepts

Trigonometric Functions

Special Angles

Rationalizing the Denominator

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