Write each function as an expression involving functions of θ&...

Question

Write each function as an expression involving functions of θ or x alone. See Example 2.

tan(180° + θ)

Accepted Answer

Hey, everyone in this problem, we're asked to write the given trigonometric expression in terms of alpha only. So we're given tangent of 360 degrees plus alpha and we have four answer choices. Option A cotangent of alpha, option B negative code tangent of alpha, option C negative tangent of alpha or option D tangent of alpha. Now, there's a couple different ways to do this. I'm gonna show you the first way, which is the quickest. OK? If you have a problem like this and you recognize this, you're gonna be able to solve this more quickly. So what we can see is that tangent of 360 degrees plus alpha is just equal to the tangent of two multiplied by 180 degrees plus F and we're just breaking that 360 degrees into two multiplied by 180 degrees. The period of tangent of X is 180 degrees. OK? So what that means is we're just adding two full periods. OK. So all we're doing is adding two full periods. Now we know that if we add a complete period, we just end up back in the same spot. OK. Adding any integer, multiple of a period gets us back to the same spot. So that means that tangent, the 360 degrees plus alpha is just going to be equal to tangent. No. Right. So we expect the answer to be D now, if you didn't pick up on that right away, you can still use trigonometric identities to simplify this. OK? So we're gonna do it that way as well so that you can see both, let's draw a line just so it's clear that we're doing a second version of this and we have tangent of something plus something. So recall that if we have tangent of A plus B, our trigonometric identities tell us that we can write this as tangent of A plus tangent of me all divided by one minus tangent of a multiplied by tangent. So what we have on the left hand side is exactly what we have. OK? We 360 degrees is going to be our A value and alpha is going to be our B value right now that we have our A val value and our B value, we can just apply the identity. So our original expression tangent of 360 degrees plus alpha, it's just going to become tangent of 360 degrees. First tangent of alpha all divided by one minus tangent of 360 degrees multiplied by tangent of alpha. Now, tangent of 360 degrees is just zero. If you didn't remember that, OK. What we want to remember is that tangent of 360 degrees is just equal to sine of 360 degrees divided by cosine of 360 degrees. And we know that sine of 360 degrees is zero, cosine of 360 degrees is 10 divided by one, just gives us zero. OK. So whether you remember the tangent value or whether you remember those separate sine and cosine values that you can use to figure out that tangent value, we still get that answer of zero. OK. So what that means is our expression becomes zero. The last tangent of alpha divided by one minus zero. OK? That's just gonna be tangent of alpha divided by one or tangent of alpha. So we get that exact same answer as we did doing it the other way. OK? So if you recognize that we're shifting by exact periods, we have a simpler way of doing it. And if not, we can use those trigger identities to simplify. Either way we find that the answer is going to be option D tangent of alpha. Thanks everyone for watching. I hope this video helped you in the next one.

Write each function as an expression involving functions of θ or x alone. See Example 2.
tan(180° + θ)

Key Concepts

Angle Addition Formula for Tangent

Tangent Function Periodicity

Reference Angles and Quadrant Sign Rules

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