Let f(x) = sin x, g(x) = cos x, and h(x) = 2x. Find the exact ...

Question

Let f(x) = sin x, g(x) = cos x, and h(x) = 2x. Find the exact value of each expression. Do not use a calculator. (h o g) (17𝜋/3)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine the exact value of F of G of 13 pi divided by six. Without the help of a calculator, we're told that F of X equals three X and G of X equals the tangent of X. Our answer choices are answer choice A nine multiplied by the squared of three answer choice B the squared of three divided by nine answer trace C, the squared of three divided by three and answer trace D the square of three. All right. So we're dealing with a composite function when we have a composite function. The second one in this case, the G is going to go inside of the F. So when we've got F of G of 13 pi divided by six, we want to start off with G of 13 pi divided by six. If G of X is the tangent of X, then G of 13 pi divided by six is the tangent of 13 pi divided by six. And before we stick that inside of the F of X, we can simplify this a bit. So 13 pi divided by six, where is that Well, if we are talking about denominators of 61 full rotation. So two pi is going to be 12 pi divided by six. 13 pi divided by six is larger than 12 pi divided by six. So we're talking about an angle that's more than one full rotation. So we can take our 13 pi divided by six and subtract a full rotation, subtract two pi which in this case is 12 pi divided by six, which will leave us with just pi divided by six. And there we go. That one's more familiar. So recalling from previous lessons, pi divided by six, that is the same as 30 degrees. And we're talking about the tangent of pi divided by six or the tangent of 30 degrees. Well, we recall from previous lessons that special right angle tangent of 30 degrees is equal to the square it of three divided by three. And if any of that was fast, definitely check out previous lesson videos on these special right angles in our unit circle. So we have got our G of 13 pi divided by six is equal to the square it of three divided by three. Now we're sticking that G of X inside of our F of X. So we want F of our G of 13 pi divided by six. We want F of G of pie divided by six. So what we're doing is taking our F of X which is three multiplied by X. So in this case, it's going to be three multiplied by our G of 13 pi divided by six, which is simplified as the square root of three divided by three. So we're just taking three multiplied by the squared of three divided by three. Well, when we do that, those threes cancel out and our F of G of 13 pi divided by six just equals the square root of three. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Let f(x) = sin x, g(x) = cos x, and h(x) = 2x. Find the exact value of each expression. Do not use a calculator. (h o g) (17𝜋/3)

Key Concepts

Function Composition

Trigonometric Values of Cosine

Periodicity and Angle Reduction

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