Using the Addition Formulas

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Question

Using the Addition Formulas

Use the addition formulas to derive the identities in Exercises 31–36.

sin (x − π/2) = −cos x

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Using the addition formulas, derive the identity for sign of 21 X minus pi divided by 2. Awesome. So it appears for this particular problem, we're asked to use the addition formulas to help us derive an identity. We're trying to derive this identity for this expression of sign of 21 X minus pi divided by 2. So now that we know that we're trying to derive this identity, let's read off our multiple choice answers to see what our final answer might be. A is cosine of 21 X, B is negative cosine of 21 X. C is sign of 21 X, and D is negative sign of 21 X. Awesome. So first off, in order to derive the identity for sign of 21 X minus pi divided by 2, using the addition formulas, we need to recall and use the sign addition formula. So let us recall and note that the sign addition formula states that sign of A plus B. is equal to sign of a multiplied by cosine of B. Plus cosine of A multiplied by sine of B. Fantastic. So, for this particular problem, let us state that A is going to be equal to 21 X and let us state that B is going to be equal to negative pi divided by 2, and let us substitute these values for A and B into our formula. And when we do that, we will find that sign of 21 X. Plus, when you take sine of 21 X plus negative pi divided by 2, fantastic, is going to be all equal to Sign of 21 X multiplied by cosine of negative pi divided by 2. Plus, cosine of 21 X. Multiplied by sign of. Negative pi divided by 2. Fantastic. So we ran out of room here, so let's move it down a little bit so we could see our equation a little bit better. Fantastic. So, once again, what we did is we substituted in our values of A and B into our expression to obtain the following. Expression. So now, moving right along here, we now need to note that, as we should recall, let's make a note of this in blue. We need to note that cosine of, once again, a cosine of negative theta is equal to cosine of theta, and we need to note that sign of negative theta is equal to negative sign of theta. Fantastic. So, with that said, we could go ahead and write that sign of 21 X plus negative pi divided by 2. It's going to be equal to sin of 21 X multiplied by cosine of pi divided by 2 minus cosine of 21 X multiplied by sin of pi divided by 2. Fantastic. So now, our next step is we need to evaluate cosine of pi divided by 2, and sine of pi divided by 2. So let us note that cosine of pi divided by 2 is equal to 0, and let us note that sine of pi divided by 2 is equal to 1, as we should recall. So now when we substitute these values back into our equation. We should obtain the following. So sign of 21 X minus pi divided by 2 is equal to sin of 21 X multiplied by 0 minus cosine of 21 X. And we're gonna take cosine of 21 X and we need to multiply it by 1. So when we simplify, we will find that sign of 21 X. So we're taking sine of 21 X minus pi divided by 2 is going to be equal to drum roll simply negative cosine of 21 X, and that is our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be multiple choice answer letter B, which once again is negative cosine of 21 X. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Using the Addition Formulas

Use the addition formulas to derive the identities in Exercises 31–36.

sin (x − π/2) = −cos x

Key Concepts

Addition Formulas

Trigonometric Identities

Co-function Identities

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