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.

sin(π + x)

Accepted Answer

Hey, everyone in this problem, we're asked to write the given trigonometric expression in terms of alpha only. So we have sine of three pi divided by two plus alpha and we're given four answer choices. Option A negative sine of alpha, option B sine of alpha, option C negative cosine of alpha and option D cosine of alpha. Now we have sign of something plus something. OK? Well, let's use our trigonometric identities to break this up into separate terms. OK. Then we'll have trigonometric functions in terms of alpha only and then we'll have trigonometric functions of three pi divided by two which will be able to simplify down to exact values. OK. All right. So recall that sign of A plus B is equal to the sine of a multiplied by cosine of B plus cosine of a multiplied by sine of be, OK. So the expression on the left hand side that we've written is exactly the same as our expression where A is equal to three pi divided by two and B is equal to alpha. OK? So let's go ahead and apply that right away. Sign of three pi divided by two whose alpha is going to be equal to S of three pi divided by two multiplied by cosine of alpha plus cosine of three pi divided by two multiplied by S of alpha. And just like we said was going to happen, we have this sine of alpha and cosine of alpha. OK. So a trigonometric function of alpha only. And then we have these sine and cosine of three pi divided by two and those we can simplify down into values. Now sine of three pi divided by two, we can think of our sine curve. If we think of the graph of sine, a sine of three pi divided by two, we have negative one. OK. So we're gonna have negative one multiplied by cosine of alpha. And then we have cosine of three pi divided by two and cosine of three pi divided by two. We can think of the curve of cosine as well, starts at one decreases and increases again and at three pi divided by two cosine is zero. So we have negative one multiplied by cos alpha plus zero multiplied by sine alpha. When we simplify that we get just negative cosine of alpha. All right. So we simplified our original expression. We've written it in terms of alpha only and we found that it's equivalent to negative cosine of alpha which matches with answer choice C. That's it for this one. Thanks everyone for watching. I hope this video helped

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

Key Concepts

Trigonometric Identities

Angle Addition Formulas

Periodic Properties of Trigonometric Functions

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