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 72°

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the exact value of the cosecant of 54 degrees. If the sine of 36 degrees equals the square root of the expression 10 minus two multiplied by the square root of five all divided by four. For our answer choices A is the square of five plus one, all divided by four. B is the spirit of five minus one C, the square root of five plus one D one minus the square root of five. Now, if we're going to find the exact value of the cosecant, and here we have a value for the sine function. We need to find some way to relate the cosecant to the sine function or some way to link their angles. Now, what do we know about the cosecant? Let's start there. So what do we know about the cosecant function? Well, when thinking about a cosecant function, we know that it's a core function to the second. In other words, recall that the course you can function is complementary to the second function. So the core second of X would be equal to the second of 90 degrees minus X. So here for the quasi count of 54 that means it's going to be the same thing as the second of 90 minus 54 degrees, which is equal to 36. Now, how does that information help us? Well, let's think, what do we know about the second? Remember we're somehow trying to link it back to the sine function. Well, we know that the S is the reciprocal of the cosine. OK. So we know that the second of an angle equals the reciprocal, in other words, one divided by the cosine of that angle. So that tells us then that the second of 36 degrees is going to be equal to one divided by the cosine of 36 degrees. Now, what do we know that relates to cosine and sine function? Well, we know the Pythagorean identity, we know that the Pythagorean identity states that the square of the sine of an angle and the square of the cosine of its angle. OK. But I said it the other way around. But you can, you know you can see them on screen here. So the square of the cosine plus the square of the sine would be equal to one. What does that tell us? Then that means that the that means then that the cosine of an angle is going to be equal to the square root of one minus the square of its sine of that angle. Now this is pretty good because now we have a value for the sine of 36 degrees. So if we can substitute our value for 36 degrees into our expression, we will be able to solve for the cosine. And then when we solve for the cosine, we will be able to find a second. And thus, that would be the value of the core. So let's go ahead by first trying to find the cosine of X. OK. So let's solve for the cosine of X. So using that identity to solve for the cosine of 36 degrees. Now, if we plug it into our formula here, that means the core sign of 36 degrees. Let me get rid of the um punctuation right there. So the cosine of 36 degrees is going to be equal to the square root of one minus the square root of the expression 10 minus two multiplied by the square root of five. OK? All divided by four, all squared. OK. Now, if we go ahead and square our term, we're fine that it simplifies one minus the square would remove the square root here in our numerator. So that would be 10 minus two multiplied by the square root of five and in our denominator four squared equals 16. Now, here we have a whole number and an expert and, and a quotient. So we can simplify that by writing them with the same denominator. OK. So you know what? Yeah, let me keep it here. So that's going to be the square root of 16. Because if we imagine this is a, with a denominator of one, both of them share a common multiple of 16, 1 multiplied by 16 equals 16. So that 16 minus or expression 10 minus two multiplied by the square root of five. And that means it's going to be 16 minus 10 if we remove the brackets minus 10 plus, because if we subtract a negative, it becomes positive. So plus two multiplied by the square root of five. So we'll have all of that in our expression, all divided by 16. We can simplify that even further because 16 minus 10 equals six. So that's six plus two multiplied by the spirit of five, all divided by 16. Now, we might be wondering here, where else can we go? Well, let's try to simplify some more. So we can find the square root of if we're finding the square root of the entire quotient, it's the same thing as finding the square root of the numerator and the denominator. So we can rewrite it as the square root of six plus two, multiplied by the square root of five, all divided by the square root of 16, which is four. OK. Now, if we have found a square root, then OK, now we need to try and simplify what's going on in our expression here because if we think about our answers, the terms were made of either or BC and D had two terms and even in a quotient had two terms. So let's think about it. Can we get five out of our expression here that will eventually help us to solve? Well, what if I split up six into five and one? So I can rewrite this here, our numerator as five plus two, multiplied by the square of five plus one. OK. And we'll have all of that divided by four. Now, if we think about it here, this is actually the perfect square because if we were to think about five as the product of the square root of five and the square root of five. So let's let me actually write that here just to show it. So if we think about five, if we rewrite it as the square of five all squared note that this is the perfect square where our first term in the binomial is the square root of five and the second term would be one. So what does that mean? Well, that means we can rewrite our expression then as the square root of five plus one, all squared under the square root all divided by four. And know that works to our advantage because the square will cancel the square root. And that is going to leave us with the square root of five plus one, all divided by four. No, we're almost there. Now, we have a value for the cosine of 36 degrees. So what does that mean? Well, no, that means we can substitute that value into what we know about the second. Remember earlier, we saw that the sea is the reciprocal of the cosine. So now this tells us, then this tells us, let me put that in blue to show that we're moving on to another, another chain of thought. That means then since the second of 36 degrees is the reciprocal of the cosine of 36 degrees is going to be equal to the reciprocal of the square of five plus one divided by all divided by four, which is four divided by the expression, the square of five plus one. No here to simplify it further, we can rationalize. And that means we're going to multiply both the numerator and the denominator by the conjugate of the spirit of five plus one which is the spirit of five minus one. Eh So no, when we do that here, OK, let's move to another line. That means we're going to have four multiplied by the square of five minus one, all divided by the square of five multiplied by the square root of five, which equals five minus OK minus one multiplied by one, which is one because remember this is the difference of two squares. In other words, recall that if we have X plus Y multiplied by X minus Y is going to be equal to the difference of those squares, X squared minus Y squared. So that's, that's where that came from. OK. So now that means here we know five minus one equals four and we can rewrite our numerator as four multiplied by the square of five minus four. OK. All divided by four. Now notice here we can break this into two fractions, four multiplied by the square of five divided by four minus four, divided by four. And our fours will cancel which leaves us with the square. Well, let me write it below it. That leaves us with a second of 36 degrees being equal to the square root of five minus four divided by four, which is one. Remember we said the sea can have 36 degrees is the same thing as the core C can have 54 degrees because it's its core function. So therefore, let me put that in red. Therefore, that tells us then that the cosecant of 54 degrees will be equal to the square of five minus one. That would be our final answer. When we look back on our answer choices that would have been answer choice. B thanks a lot for watching everyone. I hope this video helped.

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 72°

Key Concepts

Trigonometric Identities

Reciprocal Functions

Exact Values of Trigonometric Functions

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