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.

tan 72°

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the tangent of 54 degrees. If the sign of 36 degrees equals the square root of 10 minus two radical five, divided by four, our answer choices are a open parenthesis. Five plus radical five closed parentheses multiplied by open parentheses. The square root of 10 plus two radical five closed parentheses divided by 20 B, open parentheses five plus the square root of five closed parentheses multiplied by the parentheses, the square root of 10 minus two radical five closed parentheses divided by 20 C. 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 first to find the exact value of the tangent of 54 degrees. Since we're given the sign of 36 degrees. I'm first gonna think about our cotangent identities and we have one that states that the tangent of theta will equal the cotangent of 90 degrees minus theta. So in this case, we have the tangent of 54 degrees and that will be equal to the co tangent of 90 degrees minus 54 degrees, which will equal the cotangent of 36 degrees to find the cotangent of 36 degrees. Given that we have the sign, I recall that we had an identity that stated that the cosine of X is equal to the cosine of X divided by the sine of X. So since we are given sine, we need to find cosine, so we need to find the cosine of 36 degrees before we can do anything else. So one more identity to get this done, I recall that we had an identity, a Pythagorean identity that stated that the cosine of X is equal to plus or minus the square root of one minus the sine squared of X. So we have a value for the sign of 36 degrees. So let's use it in this identity. So to find the cosine of 36 degrees, we are going to use plus or minus the square root of one minus the sine squared of X. Well, we're working with 36 degrees which is in quadrant one. So this will be positive. So we don't have to worry about a negative. So we have the square root of one minus and then in parentheses, the square root of 10 minus two radical five, all of which is divided by four, close parentheses squared. So the cosine of 36 degrees equals and I'm going to put the square root back on there one minus and now squaring my parentheses when I square the numerator, I'm left with 10 minus two radical five. And when I square the denominator, I get 16, I'm gonna put this still in parentheses just because we're subtracting it. Next, I'm gonna rewrite my one as 16 divided by 16. So I have the cosine of 36 degrees equals the square root of 16 divided by 16 minus in parentheses, 10 minus two radical five divided by 16 going to the top of the next column for space. I'm gonna write these as a single fraction. So I get the cosine of 36 degrees equals the square root of 16 and make sure you distribute the negative sign into the parentheses when we rewrite this. So 16 minus 10 plus two radical five, all of which is divided by 16. Combining my like terms, I get the square root of six plus two radical five divided by 16. I'm going to take the square root of both the numerator and the denominator. So the cosine of 36 degrees equals I have the square root of six plus two radical five. And that's gonna be divided by the square of 16, which is four. There's actually one more square route we can take. And that is hiding under here, six plus two radical five actually breaks down to saying that this is the same as one plus radical five squared and it's still all divided by four. So it is hidden in there. It's not the first thing that would have come to my mind. But it is possible if I do this out, I will get six plus two radical five. So we have the cosine of 36 degrees and this will now equal and I'm gonna leave it in parentheses. One plus the square root of five divided by four. Now that we know both the sine and the cosine, we can find the cotangent of 36 degrees, which will have the same value as our tangent. So the cotangent of 36 degrees is the cosine of 36 degrees. So one plus radical five divided by four divided by the sign of 36 degrees. So the square root of 10 minus two radical five divided by four, they have the same denominator in both the numerator and denominator. So that can cancel out and that'll leave us with one plus the square root of five divided by the square root of 10 minus two radical five. And I do need to rationalize my denominator. So to do so I'm gonna multiply numerator and denominator by the square root of 10 plus two radical five because I'm using a conjugate. So I have the same terms. I just have a different middle sign going down a row, the co tangent of 36 degrees, looking at most of my answer choices. It looks like I can probably leave these in parentheses. So let's do that for now. And we'll see if that works out in the long run. So I have parentheses, one plus the square root of five multiplied by the parentheses, the square root of 10 plus two radical five. And that's gonna be divided by. When I do this out, I get the square root of 80 and the square root of 80 can be simplified into four radical five. So we'll have in parentheses, one plus radical five multiplied by the parentheses. 10 plus two radical five divided by or radical five. And we need to rationalize this one more time. I need to multiply this through once more by the square root of five. So I get the cotangent of 36 degrees equals multiplying my numerator by the square root of five. I have the square root of five times one plus the square root of five multiplied by the square root of 10 plus two radical five divided by when I multiply my denominator by the square root of five, I will be left with 20. And one last thing I want to distribute the square of five into my binomial. And when I do so I get in parentheses five plus the square of five multiplied by the square root of 10 plus two radical five, all of which is divided by 20. Yep. After all of those rationalizations, this is what we end up with. We end up with 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 that is 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.
tan 72°

Key Concepts

Exact Values of Special Angles

Trigonometric Identities

Calculator Approximations for Verification

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