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(θ - 30°)

Accepted Answer

Hey, everyone in this problem, we are asked to write the given trigonometric expression in terms of alpha only. We're given cosine of alpha minus 60 degrees. We have 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 sine of alpha divided by two. In option D cosine of alpha minus the square root of three sine of alpha divided by two, right. So what we're given is cosine and in brackets we have two angles subtracted from each other or one subtracted from the other, right? So let's think about our trigonometric identities. Is there one that matches what we've been given? OK. Can we break this up into a different expression? What we're called that cosine of A minus B is equal to cosine of a multiplied by cosine of B plus sign of A multiplied by S of B. OK. All right. What does this do for us? Well, if we look at our expression, we can see that our alpha value is going to be A and 60 degrees is going to be B, if we use this identity, we're gonna end up with cos and sine terms that just have alpha, which is good. And then we're gonna have cos and sine terms which just have 60 degrees and those we know how to simplify. OK. So this identity is going to allow us to simplify this expression. So we are gonna take A to B alpha, we are gonna take B to B 60 degrees and we are gonna apply this identity. We get that cosine of alpha minus 60 degrees is going to be equal to cosine of A, the cosine of alpha multiplied by cosine of B. So cosine of 60 degrees. What sign of a sine of alpha multiplied by sine of B which is sine of 60 degrees. So we get, we've just taken that those A terms and those B terms and we've split this thing up into their separate parts. Now let's go through and simplify. OK, cosine of alpha, we can't simplify. So that's just gonna be cosine of alpha, cosine of 60 degrees. We know what cosine of 60 degrees is cosine of 60 degrees or call is one ha. So we have cosine of alpha multiplied by one half and then we have sign of alpha again, we can't simplify that, but we can simplify s of 60 degrees s of 60 degrees or call is the square root of three divided by two. Hm. So now we have two terms, both of them have a denominator of two. So we can write these as one big term. We get cosign of alpha plus the square root of three multiplied by sine of alpha, all divided by two. OK? Just rewriting our two terms into one fraction. And that is that final expression we were looking for. OK. We now have this in terms of alpha only. Everything else has been simplified. If we compare this to our answer choices, we can see that this matches with option C Thanks everyone for watching. I hope this video helped see you in the next one.

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Write each function as an expression involving functions of θ or x alone. See Example 2.
cos(θ - 30°)