Write each expression as a sum or difference of trigonometric ...

Question

Write each expression as a sum or difference of trigonometric functions. See Example 7.

2 cos 85° sin 140°

Accepted Answer

Welcome back. I am so glad you're here. We're asked to use a product to sum formula to write the given trigonometric expression as a sum or difference. Our given trigonometric expression is two cosine 55 degrees sine 150 degrees. Our answer choices are answer choice. A sign 205 degrees minus sign 95 degrees. Answer choice BS 95 degrees minus S 205 degrees. Answer choice C cosine 205 degrees plus cosine 95 degrees and answer choice D sign +205 degrees plus S 95 degrees. All right. The product to some formula is the appropriate identity that we want to use here because we can see that we have two different trigonometric functions with two different angles being multiplied together. So this time it's a cosine being multiplied by a sign. So the product to some identity that we're going to use is when the cosine of an angle A is multiplied by the sine of a different angle B that will equal one half multiplied by the quantity of the sign of A plus B minus the sign of A minus B. And so it looks very similar to that, except we notice that in our given trigonometric expression, it's being multiplied by two. So we can get ours, our product to some identity to look like that. We're going to multiply both sides by two. And when we do that on the right hand side, one half multiplied by two. Well, that's just one. So now it looks very similar and we can identify our A and B A here, the cosine angle is 55 degrees and B here, the sine angle is 150 degrees. So now we can plug in on the right hand side. So this two cosine 55 degrees, sine 150 degrees is equal to. Now, it's no longer one half because that was canceled out by the two, it's equal to the sine of angle A is 55 degrees plus B is 150 degrees. Then minus the sign of angle A is 55 degrees minus angle B is 150 degrees. So now it's just simplifying the right hand side. So 55 degrees plus 150 degrees is 205 degrees. So we have the sign of 205 degrees minus the sign of 55 degrees minus 150 degrees is a negative 95 degrees. Now, this is very close, but we note here, this sign of negative 95 degrees sign we recall from previous lessons is an odd function which means that the sign of a negative X. Here, we'll say for the angle is equal to the negative sign of X our angle. So here the sign of negative 95 degrees is the same as a negative sign of the positive 95 degrees. But that is being subtracted from the sign of 205 degrees. So subtracting a negative is going to give us a positive. This will simplify to be the sign of 205 degrees plus the sign of 95 degrees. And there, our trigonometric function is written as a sum. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

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Write each expression as a sum or difference of trigonometric functions. See Example 7.
2 cos 85° sin 140°