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(45° + θ)

Accepted Answer

Hey, everyone in this problem, we're asked to write the given trigonometric expression in terms of alpha only. We're given sine of 30 degrees plus alpha. We have four answer choices. Option A cosine of alpha plus the square root of three multiplied by sine of alpha, all divided by two. Option B is the exact same as option A but with a minus sign instead of A plus sign in that numerator, option C sine of alpha plus the square root of three multiplied by cosine of alpha all divided by two. And option D the same as option C but with a minus sine instead of a plus sign in that numerator. So we give given sign of something plus, OK. Let's call trigonometric identities. And we know that there's one that says that sign of A B can be written as sign of a multiplied by cosine FB, what code sign of A multiplied by sine of P. So the expression on the left hand side sign of A plus B is exactly what we have. OK, where the A is 30 degrees and the B is alpha. OK? So we can write it as the big expression on the right hand side. And let's just make sure that that's gonna help us before we do all that work. If we do this, we're gonna get sine of 30 degrees which we can simplify multiplied by cosine of alpha. So that's a trigger expression just in terms of alpha like we want, then the next term we have cosine of a cosine of 30 degrees, we can simplify that. We know the value of that. And then sine of alpha, which is just going to be again, a tr Trigon meric expression in terms of alpha only. And so this is gonna get us exactly what we want. So we're gonna take the expression on the left hand side and we are gonna expand and use those A and B values. OK? So A, it is going to be 30 degrees B is going to be alpha. All right. So let's go ahead and do it. We have signed of 30 degrees plus alpha. This is going to be equal to sine of 30 degrees multiplied by cosine of alpha plus cosine of 30 degrees multiplied by sine of alpha. Now, we can go ahead and simplify sign of 30 degrees. OK? This is a value that you should know or you should be able to find using your triangles. OK? Sos of 30 degrees is going to be one half and then we have multiplied by cosine alpha plus cosine of 30 degrees. Again, use those triangles or have these values um memorized and we get cosine of 30 degrees is the square root of three divided by two. And we multiply by sign of out no, just like our answer choices have we have this two in the denominator for both of our terms. So we can write this as one whole fraction instead of these two separate terms. So we have cosine of alpha plus the square root of three multiplied by sine of alpha all divided by two. And that is that expression written in terms of alpha only. OK. Everything else is just numbers. The trigonometric functions are in terms of alpha only if we compare this to our answer choices, we can see that this corresponds with answer choice A, that's it for this one. Thanks everyone for watching. See you in the next video.

Write each function as an expression involving functions of θ or x alone. See Example 2.
sin(45° + θ)

Key Concepts

Angle Addition Formulas

Trigonometric Values of Special Angles

Function Notation and Variables

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