Evaluate each expression without using a calculator.
cos...

Question

Evaluate each expression without using a calculator.

cos (tan⁻¹ (5/12) - tan⁻¹ (3/4))

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the expression without using a calculator. Our expression is the cosine of now. This is all in parentheses. The inverse tangent of 9/40 minus the inverse tangent of 2020 firsts. Our answer choices are answer choice. A negative 611 divided by 1189. Answer choice B 789 divided by 1160. Answer choice C 660 divided by 1189. And answer choice D 1020 divided by 1189. All right. So we've got the cosine of something here and taking a closer look at that something that we're taking the cosine of. We have an inverse tangent minus another inverse tangent. And we recall from previous lessons that an inverse trigonometric function is actually an angle. So we have the cosine of an angle minus noting that that's a different angle. The inverse tangent of nine divided by 40 is not the same as the inverse tangent of 20 divided by 21. So what we can do is we can set each of those inverse tangents equal to a different angle. So we can say angle A equals the inverse tangent of nine divided by 40 because we recall from previous lessons, that's the same as saying that the tangent of A equals nine divided by 40. And then for the second one, we can set that to B, we can say that angle B is equal to the inverse tangent of 20 divided by 21. That would be the same as saying that the tangent of B equals 20 divided by 21. Why is this important? Because now we can set our inverse tangent of nine divided by 40 equal to will substitute in what it's equal to A. And then we can set our inverse tangent of 20 divided by 21 to what that's equal to which is B. And then we bring the rest down. We have the cosine of A minus B and that is a difference. And we can use a different identity to solve this here. So we recall from previous lessons, the difference identity where we have the cosine of A minus B that is equal to the cosine of a multiplied by the cosine of B plus the sine of A multiplied by the sine of B. So now we just need to figure out what the cosine of A, the cosine of B, the sine of A and the sine of B are we can plug those in because that's what our cosine of A minus B is equal to. And then we'll have this simplified. And we recall from previous lessons that when we know the tangent, we can figure out the sign and the cosine, we recall from previous lessons that the tangent is equal to the tangent of an angle is equal to Y divided by X. And so if we set our tangent of A equals equal to Y divided by X, that's equal to nine divided by 40. And for that angle, A, we can see that X equals the 40 in the denominator, our Y equals the nine and the numerator. And for our angle B, we set that one equal to Y divided by X which equals 20 divided by 21. And then our X here equals the 20 in the denominator and Y equals the excuse me, X equals the 21 in the denominator and Y equals the 20 in the numerator. And we recall that our cosine, I have an angle. We'll stick with. The coin of A is going to be equal to X divided by R. And that the sine of an angle is going to be equal to Y divided by R. And we'll set the same for angle for angle B. So the cosine of B will equal its X divided by R and the sine of B will equal its Y divided by R. We know X and Y all we need to figure out is the R of each one. So let's start with angle A, we recall from previous lessons that are equals the square root of X squared plus Y squared. So for angle A I'll put in R sub A equals the square root of X squared, X is 40 squared plus Y squared, Y is nine squared. Now, if you can do all of this part without the use of a calculator, that's impressive and very good. So I am going to take 40 squared, that is 1600 plus nine squared is 81. 1600 plus 81 is 1681. And the square rut of 1681 equals plus or minus 41. So which one is it plus or minus? Well, if we recall for a moment that we've got a coordinate plane, we've got a vertical Y axis and a horizontal X axis. They come together at the origin in the middle, our tangent for both angles A and B is equal to a positive value. And where is tangent equal to a positive value? That's in the first quadrant and the third quadrant. However, what we're talking about here is the inverse tangent for both of these, an inverse tangent has a limited range because it's a 1 to 1 function here. And so the range of our inverse tangent, we recall from previous lessons is from negative pi divided by two to positive pi divided by two. And that means we're talking about quadrants one and four only. And so we know we're not going to be in quadrant three. That one's disregarded. We're only in one and two. Since it's equal to a positive, our tangent or tangents equal to a positive, we're in quadrant one. So X and Y are both positive and that means our R is going to be positive. We can disregard the negative value here. R is only equal to positive 41 for angle A. Now, for angle B, we'll take our sub B equals the square of X is 21 that needs to be squared plus Y is 20 which needs to be squared. And so we've got the square root of 21 squared is 441 plus 20 squared is 400 441 plus 400 is 841. And the square root of 400 or excuse me, the square root of 841 is plus or minus 29. But we're just dealing with the positives here. So that's 29. So now we can just plug these in and we have our cosine of A and B and our sine of A and B. So the cosine of A is X divided by R. The X for angle A is 40 divided by R is 41. The sign for angle A Y divided by Ry for angle A is nine divided by R is 41 cosine of angle B. Our X for angle B is 21 divided by R is 29 and sign for angle by is 20 divided by R is 29. Now, we can plug all of these in and simplify, all right cosine of angle A is 40 divided by 41 multiplied by the cosine of angle B is 21 divided by 29 plus the sine of angle A is nine divided by 41 multiplied by the sine of angle B is 20 divided by 29. Now multiplying our first set of fractions, 40 divided by 41 multiplied by 21 divided by 29 gives us 840 divided by 1189 plus our second set of fractions nine divided by 41 multiplied by 20 divided by 29. Gives us 180 divided by 1189. Look at that nice common denominator. Now, we can add our numerators and that will give us 840 plus 180 is 1020 divided by our denominator 1189. That's our expression simplified. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Evaluate each expression without using a calculator.
cos (tan⁻¹ (5/12) - tan⁻¹ (3/4))

Key Concepts

Inverse Trigonometric Functions

Trigonometric Identities

Right Triangle Relationships

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