In Exercises 39–46, use a half-angle formula to find the exact...

Question

In Exercises 39–46, use a half-angle formula to find the exact value of each expression. tan(7𝝅/8)

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the exact value of the expression of the tangent of 5/6 of pi using the half angle formula. For our answer choices A is a negative square root of three B is the square root of three C is the square root of three divided by three and D is the negative square root of three divided by three. Now, we're trying to figure out the exact value using the half angle formula and recall that the half angle formula says that the tangent of a half angle equals plus or minus the square root of one minus the cosine of theta divided by one plus cosine of theta that angle. Now, how do we know if it's a positive or negative value? Well, we can use our unit circuit to help us. So let me do a little sketch of the unit circle to the right and on our unit circle, if we look at our angle in question 56 of pie, 56 of pie would land somewhere in the second quadrant over here. OK. Let me, let me draw it properly for everyone to see and at 5/6 of pi in the second quadrant, the tangent is negative. So the tangent is negative. So that means we're going to use the negative value of our half angle formula. Now, before we continue, we need to figure out what value of the we're going to use for the co sign. And we know that we're working with a tangent of 56 stuff. That means that 5/6 of pi would be equal to our half angle. So theta divided by two equal 5/6 of Pi. Therefore, to figure out what angle we're working with, we can solve for theta by multiplying both sides by two. On the left hand side, two cancers two which leaves us with theta. And on the right hand side notice that two is a factor of six. So when we factor 022, divided by two is 12, divided six, divided by two is three. Therefore, theta equals five thirds of pi know that we have the value of the, we can solve for the tangent because then that tells us that the tangent of 5/6 of pi would be equal to the negative value of the square root of one minus the cosine of five thirds of pi divided by one plus the cosine of five thirds of pi. No, we're almost there. The only problem we're running to know is that we don't know what the cosine of five thirds of pie is But again, let's use our unit circle to help us. So I'm going to do another sketch of the unit circle. OK. And on the unit circle, we are looking for some easily accessible value that's equivalent to the cosine of five thirds of pi recall that on our unit circle, each coordinate is in the form the co sign of theater and the sign of theater where the co sign of theater is the X coordinate and the sign of theater is the Y coordinate. So if we were to represent the co sign of five thirds of pi on our unit circle, it would be somewhere in our fourth quadrant because that's where the, that's where five thirds of pie would end up. OK. And at that angle, we'd have a possible X value. OK, which you can realize I've identified here, let me highlight it in yellow. But if that's the X value, if, if that value is the cosine of five thirds of pi, that means its reference would have the same X value which would fall in the first quadrant. So the question is what would be our X value at that point? Well, at that point, it would mirror five thirds of pi. If we look at five thirds of pi, it's only a third of pi away from a complete revolution. So in the opposite direction in the first quadrant, it's equivalent value would also be a third of pi away. Therefore, what does that tell us, it tells us that the cosine of five thirds of pie is equivalent to the core sign of a third of pie. And that helps because we know that the cosine of a third of pie equals a half. So since let's make a note of that, since the cosine of five thirds of pi equals or is equivalent to the cosine of a third of pi, which equals a half, then we can substitute that to solve for the tangent of 5/6 of pie. Because no, that tells us that we can write our formula as negative the negative square root of one minus the cosine of a third of pi which is a half all divided by one plus a half. No, let's go ahead and solve one minus a half equals a half and one plus a half equals three halves, a half divided by three halves. Let's work that out on. Let me put that to the right of our equation. So a half divided by three halves equals a half multiplied by two thirds. We can factor 02 and that leaves us with a third. Therefore, the tangent of 5/6 of pi equals a negative square root of one third. Now we're almost there. It means we can find the square root of both values. So that would be equal to the negative square root of one which is one divided by the negative, divided by the square root of three to simplify all we need to do now is rationalize. That means we can multiply both the numerator and the denominator by a square root of 31 multiplied by the square of three equals the square of three and the square of three multiplied by the square of three equals three. Therefore, that tells us that the tangent of 5/6 of pi equals a negative square root of three divided by three. When we look back on our answer choices, that would have been answer choice. D thanks a lot for watching everyone. I hope this video helped.

In Exercises 39–46, use a half-angle formula to find the exact value of each expression. tan(7𝝅/8)

Key Concepts

Half-Angle Formulas

Reference Angles and Angle Reduction

Sign Determination in Trigonometric Functions

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