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

Question

Evaluate each expression without using a calculator.

sin (sin⁻¹ 1/2 + tan⁻¹ (-3))

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 sign of and then a big long thing in parentheses, we have the inverse sign of 2/5 plus the inverse tangent of negative four. Our answer choices are answer choice A all in the numerator four square root 357 minus two square root 17. And that's all divided by 85 answer choice B all in the numerator negative four square root 357 plus two square root 17. That's all divided by 85. Answer choice C negative 11 divided by 85 and answer choice D three square root 17 divided by 55. All right. So you've got the sign of this big long thing and this big long thing when we look at it more closely is two inverse trigonometric functions being added together. And we recall from previous lessons that an inverse trigonometric function is just equal to an angle. So we can set that first one, the inverse sine of 2/5 equal to angle A. And we recall that the inverse sine of 2/5 equaling angle A is the same as saying that the sine of A equals 2/5. And we can set the second one, the inverse tangent of negative four equal to angle B knowing that then the tangent of angle B equals negative four. And now we can use substitution where we've got the inverse sign of 2/5 we can plug in what that's equal to which is just A and where we have the inverse tangent of negative four, we can plug in what that's equal to, which is B and bringing the rest down. Now, we've got the sign of A plus B and that is a sum and we can use some identities in order to solve here. So we recall from previous lessons that when we have the sign of A plus B that is equal to the sine of angle A multiplied by the cosine of angle B plus the cosine of angle A multiplied by the sine of angle B. So now we just have to figure out the sine of A. We actually know that one already the cosine of B, the cosine of A and the sine of B plug all of those in multiply add, simplify, we've got our exact value. So just, just a few steps. So we know the sign of angle A, we've got to figure out the cosine of angle A, we know the tangent of angle B. So we still have to figure out both the sign of B and the cosine of B. Well, we know from previous lessons that the sine function is equal to Y divided by R and that the cosine function is equal to X divided by R and the tangent function is equal to Y divided by X. And that's helpful because if we figure out our X and Y and R, then we know the missing parts can plug those in. So for our sine of angle A, that's equal to 2/5 which equals Y divided by R, so our Y value is equal to two in the numerator. And our R value is equal to five in the denominator, we just need to figure out X and then we'll know the cosine, we recall from previous lessons that R equals the square root of X squared plus Y squared. So if we've got R is five for angle A equals X there or the square root of X squared is unknown. So we plug that in as X squared plus Y squared, Y is two squared. So that will be five equals the square root of X squared plus two squared is four squaring both sides on the left, we have 25 equals on the right X squared plus four. Subtracting four from both sides, we have 21 equals X squared. Taking the square root of both sides. We have X equals plus or minus the square root of 21. But which one is it? Is it plus or minus? Well, for that, we've got to figure out where we are. So let's take a moment and draw a quick sketch of a coordinate plane. We've got a vertical Y axis, a horizontal X axis. They come together at the origin in the middle. Our sine of a equals a positive 2/5. So where is sine equal to a positive value? We recall from previous lessons that sine is positive in the 1st and 2nd quadrants. So which one are we in? Well, we're not just talking about a sine function. We're also talking about an inverse sine function and an inverse sine function has a limited range because it's a 1 to 1 function. So recalling from previous lessons and looking at the tables in the textbook, the range of a sine function is from negative pi divided by two to positive pi divided by two. Or in other words, we are limited to the 1st and 4th quadrants for our sine function, which means we can't be in the second quadrant for that positive sine value. So we have to be in the first quadrant. So in the first quadrant X values are positive, so we can disregard that negative square root 21. And we know we're just dealing with X equals a positive square root 21 for angle A, all right, we can go back to our cosine of angle A equals X divided by Rx is the square root of 21 divided by R is five. We've got our cosine of angle A one down two to go. All right, let's take a look at angle B. So we know the tangent of B equals negative four, which equals Y divided by X. Well, to get a four, we know that we'd be dividing four by one. So Y would be four and X would be one but to have a negative 41 of those is going to be negative. But which one? It could be either one. Again, we have to figure out where we are. So tangent being equal to a negative four equal to a negative value. We recall from previous lessons that tangent is going to be negative in quadrants two and in quadrant four. But since we're really talking about the inverse tangent function, we have a limited range. And so what is the range of our inverse tangent looking at those tables in the textbook? And recalling from previous lessons that is from negative pi divided by two to positive pi divided by two as well. So we're still in the 1st and 4th quadrants again, we can just forget about the second quadrant because tangent is equal to a negative value. We're in the fourth quadrant for angle B. And that means that our Y values are going to be negative here, X values are positive and the fourth quadrant Y values are negative. So now we know X is equal to positive one, Y is equal to negative four. Let's figure out R, all right. So R for angle B equals the square root of X is one squared plus Y is negative four squared. One squared is one plus negative four squared is 16, 1 plus 16 is 17. So that's the square root of 17, R equals the square root of 17. OK. Now we can plug that in. So our sine of angle B equals Y which is negative four divided by R which is the square root of 17 Y, we have a radical in our denominator, we need to rationalize that. So I'm going to do this very quickly. If you need a refresher on rationalizing a denominator, definitely check out previous lesson videos, right? If we have negative four divided by the square root of 17, we're going to multiply both the numerator and the denominator by the square root of 17. So in the numerator, we'll have negative four square root 17 divided by the square root of 17, multiplied by itself is just 17. So our sign of angle B with its denominator rationalized is going to be negative for square root 17 divided by 17. Now, for the cosine of angle B that's X divided by R and so that's one divided by the square root of 17, that needs to be rationalized again. So that will be equal to the square root of 17 divided by 17. But now we know our sign and cosign of angle B. Now we just plug all these in and simplify. All right, let's do that sine of angle A equals 2/5 multiplied by the cosine of angle B. That's the square rut of 17 divided by 17 plus the cosine of angle A. That's the square root of 21 divided by five multiplied by the sine of angle B. That one is negative four square root 17, divided by 17, multiplying 2/5 multiplied by square root. 17, divided by 17 is two square root. 17, divided by 85 plus square root 21 divided by five multiplied by negative four square root. 17 divided by 17 is negative four square root 357. If you can do all of this without using a calculator as per the instructions that is awesome and five multiplied by 17 in the denominators again, 85 we have a common denominator. That's great. We can combine these. Now, for whatever reason, the answer choices put this one first, a negative four square 357 plus two square 17. And that's all divided by 85. And that's as simplified as that one gets. That is our exact value answer. Simplified. There we go. All right, we look at our answer choices and this matches with answer choice B well done. We'll catch you on the next one.

Evaluate each expression without using a calculator.
sin (sin⁻¹ 1/2 + tan⁻¹ (-3))

Key Concepts

Inverse Trigonometric Functions

Trigonometric Addition Formulas

Right Triangle Interpretation of Inverse Tangent

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