Advanced methods of trigonometry can be used to find the follo...

Question

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

cos 18°

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the cosine of 36 degrees. If the sine of 36 degrees equals the square root of 10 minus two radical five, all of which is divided by four, our answer choices are a, the square root of five plus one divided by four B, the square root of five minus one divided by four C, the square root of five plus one divided by two D one minus the square root of five divided by four. So first thing that comes to mind here is I need to find a way to get from sign to cosign. So I recall that we have a formula or an identity that states that the cosine of X is equal to plus or minus the square root of one minus the sine squared of X. And since we are working with 36 degrees, we can just deal with the positive side of this. So we can say that the cosine of 36 degrees equals the square root of one minus our sine of 36 degrees. So the square root of 10 minus two radical five divided by four and I put that in parentheses so I can square it. So continuing to simplify the square root, I'm sorry, the cosine of 36 degrees was the square root of one minus. Now, I'm going to square both the numerator and denominator of my fraction. The square root of 10 minus two radical five squared leave us with 10 minus two radical five divided by four squared, which is 16 from here. I'm gonna rewrite one as 16 divided by 16 and combine this into one fraction. So we have the cosine of 36 degrees and it equals the square root of 16 divided by 16 minus. And I'm actually going to make sure I distribute that by putting 10 minus two radical five in parentheses divided by 16. So the cosine of 36 degrees equals the square root of 16. And then I'm gonna distribute this negative into the parentheses minus 10 plus two radical five, all of which is divided by 16. Taking another step to simplify this. I'm gonna combine the light terms that are in the numerator. So the cosine of 36 degrees equals the square root of six plus two radical five divided by 16. At this step, I'm going to rewrite this as the square root of the numerator divided by the square root of the denominator. So we have the cosine of 36 degrees equals the square root of six plus two radical five divided by the square root of 16 and continue this at the top of the next column. So this would be the cosine of 36 degrees equals the square root of six plus two radical five divided by four. So at this step, I personally would leave it because it matches my original problem. However, it does not match any of our answer choices. So we need to see, is there something else here that we aren't thinking about? And what that is is that under the radical and our original square root, we can rewrite six plus two radical five. This is actually a perfect square because if I were to have rewritten this as one plus two radical five plus five, that would have been my perfect square trinomial. So I can actually factor this under my radical to be one plus the square of five in parentheses squared, divided by four. And then, because I could take the square root of a square, those two will drop and I'll have one plus the square root of five divided by four. And that looks very similar to answer choice. A, I'll reorganize it. So it's a square to five plus one divided by four. So that way it matches our answer choice perfectly. But that is our answer answer choice. A the square root of five plus one divided by four have a nice day.

Advanced methods of trigonometry can be used to find the following exact value.
sin 18° = (√5 - 1)/4
(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.
cos 18°

Key Concepts

Exact Values of Special Angles

Pythagorean Identity

Trigonometric Identities and Sign Conventions

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