Apply the formula for cos (A − B) to the identity sin θ = cos ...

Question

Apply the formula for cos (A − B) to the identity sin θ = cos (π/2 − θ) to obtain the addition formula for sin (A + B).

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. Derive the subtraction formula for cosine of A minus B by using the angle addition formula for sign of A + B. And the cofunction identity, cosine of theta is equal to sin of pi divided by 2 minus theta. Awesome. So it appears for this particular problem, we're asked to derive the subtraction formula for cosine of a minus B. That's what we're ultimately trying to solve for, and we're asked to do this by using the angle addition formula for sine of A + B and the co-function identity. Cosine F theta is equal to sin of pi divided by 2 minus theta. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be, noting that they all state that cosine of a minus B equals some expression. So A is cosine of a multiplied by cosine of B plus sine of a multiplied by sin of B. B is 2 multiplied by cosine of a multiplied by cosine of B, plus sine of a multiplied by sin of B. C is cosine of a multiplied by cosine of B minus cosine of a multiplied by cosine of B. D is cosine of a multiplied by cosine of B plus 2 multiplied by sin of a multiplied by sin of B. Awesome. So first off, in order to derive the subtraction formula for cosine of a minus B using the angle addition formula for sine of A plus B and the identity, cosine. theta is equal to sin of pi divided by 2 minus theta. We need to follow these few steps. So our first step is we need to, as we should recall a note, we need to start with the angle addition formula for sign. So let us recall that the angle addition formula for sign states that sign of A plus B. is equal to sign of a multiplied by cosine of B. Plus, cosine of A multiplied by sign of B. Fantastic. So now our second step is we need to use the identity cosine of theta is equal to sin of pi divided to minus theta to express cosine in terms of sign, so meaning we need to write that cosine of A minus B is equal to sign. Of pi divided by 2 minus A minus B. Fantastic. Our third step is we need to rewrite pi divided by 2 minus A minus B as pi divided by 2 minus A plus B. So when we do that, we'll be able to write that cosine of A. Minus B will be equal to sign of. I divided by 2 minus A plus B. Fantastic. Our 4th step is we need to apply, rather, we need to recall and apply the angle addition formula for sign. So when we do that, we'll be able to write that sign of I divided by 2 minus A plus B. So, plus B, will be equal to sign of. I divided by 2 minus A multiplied by cosine of B. Plus co-sign of. I divided by 2 minus A. And then it's gonna be multiplied by sign of B. Fantastic. So, we ran out of room for a second here, so let's move it over just a, just a little bit. So it's gonna be multiplied by sign of B. Fantastic, so sign of B. Fantastic. So now, moving right along here, so our 5th step now is we need to use the cofunction identity again in order to help us simplify. So we need to write that sign of pi divided by 2 minus A, it's going to be equal to cosine. Of a And Cosine of pi divided by 2 minus A is gonna be equal to sign of A. Fantastic. So now, moving right along here, our 6th step is we need to substitute these values back into our equation. So we need to say that cosine of A minus B is gonna be equal to cosine. Of A multiplied by cosine of B. Plus sign of a. Multiplied by sine of B. Fantastic. So, therefore, without a doubt, this is the subtraction formula for cosine of A minus B. So our final answer. Without a doubt, as to be once again, cosine of a minus B is equal to cosine of a multiplied by cosine of B plus sin of a multiplied by sin of B, and this corresponds with multiple choice answer letter A, which once again. Cosine of A minus B is equal to cosine of a multiplied by cosine of B plus sine of a multiplied by sin of B. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Apply the formula for cos (A − B) to the identity sin θ = cos (π/2 − θ) to obtain the addition formula for sin (A + B).

Key Concepts

Trigonometric Identities

Cosine of a Difference Formula

Sine Addition Formula

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