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.

cos(45° - θ)

Accepted Answer

Hey, everyone in this problem, we're asked to write the given trigonometric expression in terms of alpha only. We have cosine of 60 degrees minus alpha. We're given four answer choices. Option A cosine of alpha plus sine of alpha divided by two. Option B cosine of alpha minus sine of alpha divided by two option C cosine of alpha plus the square root of three multiplied by sine of alpha, all divided by two. In option D cosine of alpha minus the squared of three multiplied by sine of alpha all divided by two. OK. So we have cosign of 60 degrees minus alpha. Now recall that cosine of A minus BK our trigonometric identity for the difference when we're dealing with cosine can be written as cosine of a multiplied by cosin ay, what sign of a multiplied by S of B? OK. So the left hand side is exactly what we have and we can represent our expression as the left hand side where 60 degrees is A and alpha is B. OK. So we have A is equal to 60 degrees and B is equal to alpha. Hey, this is gonna allow us to break this expression we were given into separate expressions. We're gonna get cosine and sine of 60 degrees which we can evaluate exactly. And then we're just gonna have cosine and sine of alpha. And that alpha term is gonna be all by itself like we want. OK. All right. So let's go ahead and apply that cosine of 60 degrees minus alpha is therefore going to be equal to cosine of 60 degrees multiplied by cosine of alpha, yes, fine of 60 degrees multiplied by sign. Who know now cosine of 60 degrees recall that this is equal to one half. OK. These are values that you're gonna want to remember or to be able to figure out quickly. OK? Using either your identity triangles or the unit circle. So cosine of 60 degrees is one half. So we get one half multiplied by cosine alpha plus sine of 60 degrees, which is gonna be the square root of three divided by two multiplied by sine of alpha. Now, we have a denominator of two in each of our terms. So we can simplify this and write it as a single fraction. So we get cosine of alpha plus the square root of three multiplied by sine of alpha, all divided by two, right. And that is a way of writing this original expression that we were given so that the trigonometric functions are in terms of alpha only. OK. All right. So comparing what we found to the answer choices we can see that option C is a correct answer. Cosine of alpha plus the square root of three sine alpha all divided by two. Thanks everyone for watching. I hope this video helped see you in the next one.

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

Key Concepts

Trigonometric Identities

Angle Measurement

Function Composition

Watch next