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.

cot 18°

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the cotangent of 36 degrees. If the sine of 36 degrees equals the square root of 10 minus two radical five divided by four. Our answer choices are a, the parentheses five plus the square root of five multiplied by the parentheses, the square root of 10 plus two radical five, all of which is divided by 20 B. The parentheses five plus the square root of five multiplied by the parentheses, the square root of 10 minus two radical five, all of which is divided by 20. See the parentheses, one plus the square root of five multiplied by the parentheses, the square root of 10 plus two radical five, all of which is divided by 20 D. The parentheses five plus the square root of five multiplied by the parentheses, the square root of 10 plus two radical five, all of which is divided by 80. OK. We have a little ways to go to get there, but we'll get there. First thing I noticed is our ultimate answer wants a cotangent. So I recall that the cotangent of X is the same as the cosine of X divided by the sine of X. So in this case, the cotangent of 36 degrees will be the cosine of 36 degrees divided by the sine of 36 degrees. We have the sign of 36 degrees. We need to find the cosine. So thinking of other identities, I recall that the cosine of X equals plus or minus the square root of one minus the sine squared of X. So that's what we're going to use. So the cosine of 36 degrees will equal and we don't have to worry about the negative side because 36 degrees is in quadrant one and cosines positive there, but it will equal the square root of one minus the sine of 36 degrees, which is the square root of 10 minus two radical five divided by four in parentheses squared. So let's do out that square, our cosine of 36 degrees will equal the square root of one minus. And I'm gonna leave my answer in parentheses. So it helps me with my next step. My numerator of the square root of 10 minus two radical five squared will leave me with 10 minus two radical five. The denominator of four squared is 16. I'm now gonna treat the one as if it's 16, divided by 16. So I can combine my like terms and have one single fraction. So my common denominator here will be 16 and I'll have 16 minus I'm distributing the negative into the parentheses. So minus 10 plus two radical five. So next, combining the like terms in my numerator, I get the square root of six plus two radical five divided by 16. And I want to be able to simplify this a little further. So I'm going to take the square root of both the numerator and the denominator. And I get the square root of six plus two radical five divided by four. One last thing here, I actually can simplify this a little bit more because six plus two radical five is actually a binomial square. It is the same as saying one plus the square root of five squared. So if I rewrite this as the square root of one plus the square root of five squared divided by four, I can take the square root of my square. And I'm left with the cosine of 36 degrees equals one plus radical five divided by four. So now that I know the cosine of 36 degrees and the sign of 36 degrees, we can work towards the cotangent of 36 degrees. So again, our cotangent of 36 degrees has to be the cosine of 36 degrees divided by the sine. So our cosine of 36 degrees is one plus the square of five divided by four. And that'll be divided by our sine of 36 degrees, which is the square root of 10 minus two radical five divided by four. Since in our complex fraction, our numerator fraction and denominator fraction share a denominator. We can reduce that right off the bat. So I can rewrite this as one plus the square root of five divided by the square root of 10 minus two radical five. Well, I definitely need to rationalize that denominator because there's two radicals in there already. So to do so I'm going to keep it under the radical. So I'll keep a square root, but I am gonna use the conjugate of the binomial that's under it. So we'll multiply numerator and denominator by the square root of 10 plus two radical five. Looking at my answer choices, I think it's safe to not foil out the numerator and just put it in parentheses multiplied next to each other. So I'm gonna rewrite this as the cotangent of 36 degrees equals in parentheses, one plus the square root of five multiplied by the square root of 10 plus two radical five. And in my denominator, it'll still be under a radical for now because a radical multiplied by a radical is still a radical. But when I do the product of a sum in a difference under our radical, we get 100 minus 20 which is 80. So we will have the cotangent of 36 degrees equals the numerator of parentheses. One plus the square root of five multiplied by the parentheses, the square root of 10 plus two radical five. And in my denominator, I'll have the square root of 80 which can be simplified because 80 is 16 multiplied by five. So this would actually be four radical five. So I have one more step to do, I need to rationalize this denominator again. So to do that, I will be multiplying the numerator and denominator by the square root of five. So I get that the cotangent of 36 degrees equals recall when you multiply that when we have three values multiplied together, I can multiply it by either factor. So I'm going to do the square to five multiplied by one, which will give me the square to five and then the square to five multiplied by the square to five, just five. And I'm not going to touch the messier radical. So the square root of 10 plus two radical five and then in my denominator or multiplied by the square to five, multiplied by the square to five is 20. So believe it or not, we got all the way to our answer. So my numbers are in a slightly different order in my first set of parentheses. But I found that the cotangent of 36 degrees equals the parentheses, the squared of five plus five multiplied by the parentheses, the square root of 10 plus two radical five divided by 20. And that's answer choice. A 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.
cot 18°

Key Concepts

Exact Values of Special Angles

Trigonometric Identities

Using Algebraic Manipulation to Derive Exact Values

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