Work each problem.
Find the exact values of sin x, cos x...

Question

Work each problem.

Find the exact values of sin x, cos x, and tan x, for x = π/12 , using

a. difference identities

b. half-angle identities.

Accepted Answer

Hello, everyone. We are asked to find the exact value of the sine of theta, cosine, theta and tangent of theta by using different identities or half angle identities. When theta is five pi divided by 12, we are given four answer choices. All of which have a value for sine of five pi divided by 12 cosine of five pi divided by 12 and tangent of five pi divided by 12. But the values are slightly different to begin with. I've decided we are gonna use difference identities instead of half angle identities. It seems to work out cleaner. So thinking about theta as five pi divided by 12, I want to rate this as the difference of two values. So I'm thinking that five pi divided by 12, it is equivalent to eight pie divided by 12 minus three pi divided by 12. And if I reduce those, I get two pi divided by three minus pi divided by four. And I recognize two pi divided by three and pi divided by four as values on my unit circle. So that'll help us. So for the remainder of this problem angle A will be two pi divided by three and angle B will be pi divided by four. So anywhere we need that in an identity, that's what I'm going to use. So starting with the sign, so the difference formula or difference identity for sine is the sign of A minus B equals the sine of A multiplied by the cosine of B minus the cosine of A multiplied by the sine of B. So let's plug in the values we know we are looking for the sine of five pi divided by 12. So it's going to be the sine of A which is two pi divided by three, multiplied by the cosine of B. So pi divided by four minus the cosine of two pi divided by three multiplied by the sine of pi divided by four. So now using a unit circle, I'm going to find the exact values for those functions. So the sine of two pi divided by three is the square root of three divided by two. We're gonna multiply that by the cosine of pi divided by four, which is the square root of two divided by two, subtracting the cosine of two pi divided by three, which is negative one half multiplied by the sine of pi divided by four, which is the square root of two divided by two, doing out our multiplication. Our first set of fractions, we get the square root of six divided by four, second set of fractions, we are subtracting it and then we have negative square root of two divided by four, double negatives are like adding. So we can rewrite this as a single fraction as they already have the same denominator. So the sine of five pi divided by 12 will equal the square root of six plus the square root of two, all of which is divided by four. So we have our first part done, we have the sign next, we need to find the cosine. So moving up to the top, I recall the difference. Identity for cosine is the cosine of A minus B equals the cosine of a multiplied by the cosine of B plus the sine of A multiplied by the sine of B. So again A and B, we're going to use the same values. So it's gonna be the cosine of five pi divided by 12 equals the cosine of two pi divided by three, multiplied by the cosine of pi divided by four plus the sine of two pi divided by three, multiplied by the sine of pi divided by four. So just like our previous identity, we are going to use our unit circle to find those values. So the cosine of two pi divided by three is negative one half multiplied by the cosine of pi divided by four, which is the square root of two divided by two. And then we're adding the sine of two pi divided by three, which is the square root of three divided by two. And we are multiplying that by the sine of pi divided by four, which was also the square root of two divided by two. So now doing out my multiplication cosine of five pi divided by 12 will have negative square to two divided by four plus the square root of six divided by four. And I'm going to rewrite that over one single fraction since they already have a common denominator of four. And I'm reorganizing it just so my negative isn't first. So I'm gonna put the square to six minus the square to two divided by four. So our cosine of five pi divided by 12 is the square to six minus the square to two divided by four. All right almost there. For a tangent, our difference formula is the tangent of A minus B equals the tangent of A minus the tangent of B divided by one plus the tangent of a tangent of B. So one little snag here is on our unit circle. We can find the sign and cosine, but we don't know what the tangents are. So let's first find that I'm gonna do that right in this corner, the tangent of A. So the tangent of two pie divided by three will be the sine of two pi divided by three, divided by the cosine of two pi divided by three. So the sine of two pi divided by three was the square to three, divided by two the cosine of two pi divided by three was negative one half in our complex fraction cause they have the same denominator I can cancel that out. And this will tell me that this will equal the negative squared to three. So the tangent of A is negative radical three doing the same with the tangent of B. So the tangent of pi divided by four, it would be the sine of pi divided by four, divided by the cosine of pi divided by four. The sine of pi divided by four is the square root of two divided by two. The cosine of pi divided by four is also the squared of two divided by two. So that equals one. So our tangent of B equals one knowing those values is gonna make this so much simpler for us and I'm gonna just box that off. So it doesn't get blended in with the rest of the problem. And now I can plug this into this identity. So the tangent of A was negative radical three minus the tangent of B which was one divided by one plus the tangent of A which was negative radical three multiplied by the tangent of B which was one. So right now I have negative radical three minus one in the numerator. And in my denominator, I have one minus the square root of three. I want to rationalize the denominator. So I'm going to multiply it by the conjugate. So that would be one plus the square root of three multiplying both top and bottom. So I have to do a little bit of foiling and that's fine in the numerator burst gives me negative radical three outer, gives me a negative three inner negative one. And last would be negative radical three. In the denominator, we're rationalizing it. So we purposely have the difference of or the product of a sum and a difference. So we square, the first term one squared as one square, the last term squared of three squared is three and it will be a minus in between them. So let's combine our like terms if I combine the numbers before the square roots, I have negative four minus two radical three. And that's all divided by negative two. And since each term in this fraction could be reduced by a negative two, this will reduce to two plus the square root of three. And that is our tangent of A minus B which is our tangent of five pi divided by 12. So that is the last piece to our answer. So now looking at my answer choices, I need one that has six, the square to six plus the square to two divided by four for the sign the square to six minus the square to two, divided by four for the cosine and two plus the square to three for the tangent. And that looks like it's answer choice A have a good Day.

Work each problem.
Find the exact values of sin x, cos x, and tan x, for x = π/12 , using

a. difference identities

b. half-angle identities.

Key Concepts

Difference Identities

Half-Angle Identities

Exact Values of Trigonometric Functions

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