Use one or more of the six sum and difference identities to solve...

Question

Use one or more of the six sum and difference identities to solve Exercises 13–54.

In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.

sin 40° cos 20° + cos 40° sin 20°

Accepted Answer

Welcome back. I am so glad you're here. We are asked to express the following sum as one trigonometric function only then determine its exact value using a unit circle. Our sum is the sign of degrees multiplied by the cosine of degrees plus the cosine of degrees multiplied by the sign of 82 degrees. I'm sorry. I'm not sure why that's crooked as I wrote it. All right. Our answer choices are answer choice. A, the sine of 14 degrees and the squared of two divided by two answer choice B the sign of degrees and one half answer choice C the cosine of 14 degrees and one half and answer choice. D the cosine of 150 degrees and the negative squared of three divided by two. All right. So what we've got here, we could notice there's a little bit of a pattern here, we have the sign of 68 degrees being multiplied by the cosine of 82 degrees. And then that's being added to there's 68 degrees again, but this time, the cosine of 68 degrees multiplied by, there's 82 again, but this time, it's the sign of 82 degrees. So we could say that 68 degrees can be angle A and degrees can be angle B and we could rewrite this as taking the sign of A multiplied by the cosine of B plus the cosine of A multiplied by the sign of B. And hey, that's familiar, we recognize that that is the sign of open parentheses A plus B. And since we know what A is 68 degrees, and we know what B is, it's 82 degrees, we can add those together. So taking the sign of, instead of A, we've got 68 degrees plus, instead of B, we've got 82 degrees, adding those two together, that's 150 degrees. So we're taking the sign of degrees. And that is one of our answers. That's the one trigonometric function. Now let's figure out its exact value and we get to use a unit circle. I'm going to draw a really, really rough unit circle. We've got a horizontal X axis, vertical y axis, they come together at the origin in the middle. I'm going to overlay that with what almost resembles a unit circle, putting three hash marks in each quadrant on the circle. And recalling from previous lessons, that 150 degrees that's in quadrant two, that is the hashmark that is closer to the negative part of the X axis than the positive part of the y axis. There are 150 degrees. And again, if you need a quick refresher on unit circles, definitely check out previous lesson videos. What's our reference angle for 150 degrees? That's going to be 30 degrees. So looking at the sign of 30 degrees, we can recall our special triangle with 30 degrees, 60 degrees and 90 degrees. And and we recall that the hypotenuse here is going to be two, the adjacent sign side is going to be the square of three and the opposite side is going to be one. So for the sign, we're taking the opposite over the hypotenuse. So for the sign of 30 degrees opposite here is one divided by the hypotenuse. Here is two last question. Very important. We are in quadrant two is sign positive or negative in quadrant two and sign is positive and quadrant two. So this is going to remain a positive one half. All right. So we've got our two parts of our answer. We look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. sin 40° cos 20° + cos 40° sin 20°

Key Concepts

Sum and Difference Identities

Sine and Cosine Functions

Exact Values of Trigonometric Functions

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