In Exercises 33–42, let sin t = a, cos t = b, and tan t = c.

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Question

In Exercises 33–42, let sin t = a, cos t = b, and tan t = c.

Write each expression in terms of a, b, and c.
cos t + cos(t + 1000𝜋) - tan t - tan(t + 999𝜋) - sin t + 4 sin(t - 1000𝜋)

Accepted Answer

Hello, everyone. We are asked to rewrite the following expression in terms of H K and M, if the sign of A equals H, the cosine of A equals K and the tangent of A equals M, the expression we are given is negative sign of A plus five sign of in parentheses. A minus 2200 pi closed parentheses minus the cosine of A minus the cosine of in parentheses. A plus 800 pi closed parentheses plus the tangent of A minus five tangent of in parentheses. A plus 767 pi closed parentheses. Our answer choices are a negative H minus two K plus four M B negative H plus two K plus four M C 4h minus two K plus four M D 4h minus two K minus four M. So we're gonna look at this one function at a time. So I'm gonna start by looking at sign. So we have negative sign of A plus the five sign of in parentheses. A minus 220 pi closed parentheses. So we know that the sign of A we're gonna call H so we can quickly replace the first term as negative H because negative sign of A is very straightforward, we do need to think about what the period of sign is. So the period of sign is pi. So here if we think about a minus 200 or sorry, 2200 pi pie is the same as ele uh 1100 full rotations. So if I do a minus 2200 pie, it will still equal the sign of A. So we can say that five sign of a minus 2200 pi is the same as saying the sign of A. So five sign of A would be like saying plus five H. Next I would think about the cosine. So I have negative cosine of A minus the cosine of in parentheses. A plus 800 pi closed parentheses. So we know the cosine of A is gonna equal to K and we know that the period of cosine is chi pie. So if I look at a plus 800 pi 800 pie, if we divided that by two pie would mean that this angle just makes 400 full rotations and ends at the same spot. So saying the cosine of A plus 800 pi would be the same as the cosine of A. So if I rewrite it this way, I have the cosign of A minus the cosine of A and substituting that in for K, I have negative K minus K. I'm thinking about tangent. We are given the tangent of a minus five tangent of A plus pi in parentheses, we want the tangent of A to be equal to M and we recall that the period of tangent is pie. So A plus 767 P means that we are doing 767 rotations around our unit circle if you will. And that will mean that the tangent of A plus 767 P is equivalent to the tangent of A. So if I plug this in, I'll have the tangent of A minus five, multiplied by the tangent of A. And when I substitute M in, I now have M minus five M. So now that I've broken down each piece, I'm gonna to simplify them. So for sine, I had H plus five H which will give me 4h for K I K minus K, I'm sorry, negative K minus K. So there'd be negative two K and for M, I have M minus five M. So that's negative four M. If I put those all together, I have 4h minus two K minus four M and that matches answer choice. D have a nice day.

In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c. cos t + cos(t + 1000𝜋) - tan t - tan(t + 999𝜋) - sin t + 4 sin(t - 1000𝜋)

Key Concepts

Periodic Functions

Trigonometric Identities

Angle Transformation

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