Simplify each expression. See Example 4.
⅛ sin 29.5° cos...

Question

Simplify each expression. See Example 4.

⅛ sin 29.5° cos 29.5°

Accepted Answer

Welcome back. I am so glad you're here. We're asked to simplify the following trigonometric expression. We are given 1/16 multiplied by the sign of 33.5 degrees multiplied by the cosine of 33.5 degrees. Our answer choices are answer choice. A 1 32nd S 67 degrees. Answer choice B 1/16 sign 67 degrees, answer trace C 1/8 sign 67 degrees and answer choice. D 1 64th sign 67 degrees. All right. So first we take a look at our trigonometric expression and see if it looks like any of our identities. And this looks like the double angle identity where we have two multiplied by the sine of an angle. We'll call it a multiplied by the cosine of that same angle A and that is equal to the sine of two A. Here, we can see that our angle A is 33.5 degrees. And so we just need to make our double angle identity look a little bit more like our given expression. So what's similar between them? Well, it's both a constant term being multiplied by the sine of an angle being multiplied by the cosine of an angle. And what's different is the constant term in front. So how can we make our double angle identity's constant term look like the one in our given expression? Well, if we take two and we divide it by 32 and that's going to give us 1/16. So what we're going to do is multiply or excuse me, divide both sides of our equation and our double angle identity by 32. So now on the left, instead of a two, that's two divided by 32 that's 1/16 multiplied by the sine of angle. A multiplied by the cosine of angle A there that looks like our trigonometric expression equals. And then on the right, that's going to be 1 32nd multiplied by the sine of two A. Now we can go back through and substitute in for angle A in these three places where it occurs the 33.5 degrees. So we'll have 1/16 sine 33.5 degrees, cosine 33.5 degrees equals 1, 30 seconds of two multiplied by 33.5 degrees. And now the left hand side looks exactly like what we were given. So we know that's equal to what's on the right. And we just need to simplify what's on the right hand side. So we'll keep the 1 32nd as it is. And then we'll keep this, the sign of now we'll just multiply two multiplied by 33.5 degrees is 67 degrees. And there is our answer 1 32nd sign 67 degrees. We look at our answer choices and that matches with answer choice. A well done. We'll catch you on the next one.

Simplify each expression. See Example 4.
⅛ sin 29.5° cos 29.5°

Key Concepts

Trigonometric Identities

Sine and Cosine Functions

Angle Measurement

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