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.

sin 162°

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine the exact value of the sign of 144 degrees. If the sign of 36 degrees equals in the numerator, the square of 10 minus two square at five, that's all divided by four. Our answer choices are answer choice. A in the numerator, the square root of 10 plus two square root five, all divided by four. Answer choice B the square root of 10 plus two square root five. Answer choice C in the numerator, the square root of two square root five minus 10, all divided by four. And answer choice D in the numerator, the square root of 10 minus two square root five, all divided by four. All right. So we've got the sign of two different angles. And what you notice if you play around with this long enough is that if you take 144 plus 36 both of them degrees, that's going to equal 180 degrees. And now this can be really helpful because if we took 180 degrees minus the 144 degrees, that's the 36 degrees. And that is a difference. We're taking one angle minus another angle and we have something that we know from that. So here is what we can do. We can use the difference identity for sign. We recall from previous lessons that the difference identity for sign is that 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. And here's where that 180 comes in handy. What we can do is we can set our angle A equal to 180 degrees. And then we can, for now, we'll just keep our angle B as an X. So let's go through and plug those back in. So we've got the sign of open parenthesis. A is 180 degrees minus RB is X equals the sine of A is 180 degrees multiplied by the cosine of B is X. And then that's going to be minus the cosine of A is 180 degrees multiplied by the sine of B is X and the sine of 180 degrees, that's 00 is being multiplied by the cosine of X. Then minus the cosine of 180 degrees is a negative one and that's multiplied by the sine of X. So here zero multiplied by anything. But here specifically the cosine of X that's zero and then minus negative one is going to be plus one. So it's just equal to the sine of X when we take the s of 180 degrees minus anything specifically X here. So if we set our X equal to what we were subtracting from the 180 degrees, we'll set X equal to 144 degrees because we're looking for the sign of 144 degrees. That's what we want. If we take the sign of 180 degrees minus our X is 144 degrees. 180 degrees minus 144 degrees is as we know, 36 degrees. So the sign of 36 degrees is equal to the sign of 144 degrees. And we know what the sign of 36 degrees is, that is the square root of 10 minus two square root five all divided by four. And so that's true for both the sign of 36 degrees and the sign of 144 degrees, we look at our answer choices and that matches with answer choice. D 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.
sin 162°

Key Concepts

Trigonometric Identities

Reference Angles

Calculator Approximations

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