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.

csc 18°

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the cosecant of 36 degrees. If the sign of 36 degrees equals in the numerator, this is all under the radical, the square rut of 10 minus two square root five. And then that's divided by, this is not underneath the radical four. Our answer choices are answer choice A in the numerator all under the radical, the square root of 50 plus 10 square root five. And that's divided by not underneath the radical. Just five answer trace B very similar structure. In the numerator, we have the square root of 50 minus 10 square root five. And that's all divided by five. Answer choice C similarly in the numerator, the square rut of 50 plus two square root five, that's all divided by five. And answer choice D again, similar in the numerator. It's the square root of 50 plus 10 square at five and that's all divided by four. All right. So we recall from previous lessons that the Koc can't function is the reciprocal function of the sine function. So the Koi Can't function is one divided by the sine function here, it's the Koi Cant is 36 degrees, sine is 36 degrees. So really, we, because we have this equal to an exact value for a fraction, all we need to do is flip the fraction over. And you might think like, hey, this is the fastest I've ever solved a problem. So the K can of 36 degrees equals we just flip over what the sign of 36 degrees is the four goes in the numerator. And that's divided by, in the denominator, the square root of 10 minus two square root five. And indeed, that would be like land speed record solving a problem. However, we want to rationalize this denominator. So we're going to get rid of this square root five in here first by multiplying by the conjugate. So we're going to multiply both the numerator and the denominator by square root of 10 plus two square root five. And so we'll be foiling out that denominator the 1st 10 multiplied by 10. And this is all underneath the radical 10, multiplied by 10 is 100. And then the outsides and the insides cancel each other out by design. And then our lasts negative two square root five multiplied by a positive two square root five is going to be minus four square root 25. We'll simplify that on the next step. All right, in our numerator, we have four multiplied by the square root of 10 plus two square root five. OK. We'll simplify where we can. So in our denominator underneath the radical negative four, multiplied by the square root of 25 is negative four and multiplied by five, negative four, multiplied by five is 20. So we have underneath the radical 100 minus 20 that is 80. So we've got the square root of 80 in our denominator, we've still got a radical in the denominator. We'll have to rationalize again in the numerator is four square root 10 plus two square root five. So let's rationalize by multiplying both the numerator and the denominator by the square root of 80. Now, in the numerator, we're taking four multiplied by the square root of 10 plus two square root five, multiplied by the square root of 80. We can note here that 80 is 16, multiplied by five and 16. We can take the square root of that. That's four. So the square root of 80 is the same as four square root five. So we can take this four square root 10 plus two square root five and multiply that by four square root five. So multiplying our fours in front, that's going to be a 16, then square root and then inside the radical will have that five multiplied by the quantity of 10 plus two square root five. That's all divided by the square root of 80 multiplied by itself is 80 and now 16 and 80 reduce 16 goes into 85 times. So that's 1/5. And inside of our radical, we can distribute the five to the 10 plus two square root five. So this will be in the numerator, the square root of five multiplied by 10 is 50 plus five, multiplied by two is 10 square at five all divided by. There's five left in the denominator. And well, there we go, there's the cosecant of 36 degrees. Just a little bit of a pain because we rationalize the denominator. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

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.
csc 18°

Key Concepts

Trigonometric Functions

Reciprocal Identities

Exact Values and Approximations

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