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 (A − B) = sin A cos B − cos A sin 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. Using the sine addition formula, solve for sign of 9 x minus 5Y. So that's it. What we're ultimately trying to solve for is we're trying to take the sign of 9 X minus 5Y, and we're trying to solve it using the sign addition formula. 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. A is sin of 5 x multiplied by cosine of 9 Y plus cosine of 9 X multiplied by sin of 5Y. B is sin of 5 X multiplied by cosine of 9 Y minus cosine of 9 X multiplied by sin of 5Y. C is sign of 9 X multiplied by cosine of 5 Y minus cosine of 9 X multiplied by sin of 5Y. And D is sign of 9 X multiplied by cosine of 5 Y plus cosine of 9 X multiplied by sin of 5Y. Awesome. So like the problem itself states, we need to use the sign addition formula. So first off, let us recall and use the signedition formula. So as we should recall, The sine addition formula states that sign of A plus B, so we're trying to say, once again, the sine addition formula states that sign of A plus B is going to be equal to drumroll, sign of A. Multiplied by cosine of B plus cosine of a multiplied by sign of B. Fantastic. So now let us define our variables. So let us state that A is going to be equal to 9 X and let us say that B is going to be equal to -5Y. Fantastic. So now when we plug in our different values for A and B into our expression, we will be able to simply write that. Sign of 9 X minus 5, so sign of 9 x minus 5Y is going to be equal to sign of 9 X multiplied by cosine of negative 5Y. Plus cosine of 9 X multiplied by sine of -5Y. Fantastic. So now our next step is we need to recall and use trigonometric identities to help us simplify our expression. So let us recall and use even odd properties. So let us recall that Cosine of negative theta, where theta is some angle, so cosine of negative theta is equal to cosine theta, which means that cosine is even, I'll say E for even. And let's state and recall and remember that we could state that sign of negative theta is equal to negative sign of theta, which means that sign is an odd function, which we'll call O for short. So now we need to apply these properties to our cosine of -5Y and our sign of negative 5Y. So let us note that cosine of -5Y is equal to cosine of 5Y. fantastic. And let us also note that Sign of -5Y is going to be equal to negative sign of 5Y. So now, when we substitute these values back into our expression, we will find that we will get of 9 X minus 5 Y is equal to. Sine of 9 X multiplied by cosine of 5 Y. Plus cosine of 9 X. Minus sign of 5 Y. So when we simplify the second term, we will be able to state that sign of 9 X minus 5Y it's going to be equal to sign of 9X multiplied by cosine of 5Y. Minus cosine of 9 X multiplied by sin of 5 Y. And that's it. We've officially solved for this problem. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be multiple choice answer letter C, which once again is sign of 9 X multiplied by cosine of 5 Y minus cosine of 9 X multiplied by sign of 5Y. 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 (A − B) = sin A cos B − cos A sin B

Key Concepts

Addition Formulas

Trigonometric Identities

Derivation Techniques

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