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

Question

Evaluate each expression without using a calculator.

sin (2 tan⁻¹ (12/5))

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 two inverse tangent of 35 divided by 12. Our answer choices are answer choice. A 925 divided by 1276. Answer choice B negative 311 divided by 849. Answer choice C 840 divided by 1369. And answer choice D 226 divided by 389. All right. So we've got this expression here and in the middle, in the middle of it, we've got an inverse trigonometric function which means that this part where we've got the inverse tangent of 35 divided by 12. What we really have there is we have the angle. So we can say that's angle theta. And really what it is is when we have the tangent of an angle, theta, it would be equal to 35 divided by 12. And that can be helpful. We can just call that angle theta for now. And what we have then is the sign of two theta. And when we have the sign of two multiplied by an angle, that's a double angle. And we can use a double angle identity. We know our signs, double angle identity. We've got the sign of from the tables in the textbook. And from previous lessons, we'll say the sign of two A equals two S A cosine A and here our A would be angle theta. So this will be simplifying two, ours of two, theta will be two multiplied by the sine of theta, multiplied by the cosine of theta. So what are the sine of theta and the cosine of theta? Well, if we recall from previous lessons, the tangent of an angle is equal to Y divided by X. So we can see here looking at our 35 divided by 12 equals Y divided by X, we've got an X value of 12 and A Y value of 35. Now, a couple things to note, we're talking about inverse tangent which gives us a limited range. We recall from previous lessons that the range of inverse tangent is going to be from negative pi divided by two to positive pi divided by two. Or in other words, we are in the first and fourth quadrants. And so we've got a positive Y value and a positive X value we are in the first quadrant here. So we're just dealing with positive values. Another thing to note is that we recall from previous lessons that the cosine of an angle theta is going to equal X divided by R and the sign of an angle theta is going to equal Y divided by R. Well, we know our X and our Y, we just need to figure out R. We also recall that R equals the square root of X squared plus Y squared. Now, if you can do this part without a calculator as according to the instructions, more power to you. So we're plugging in the square root of X, as we said is 12, that'll be 12 squared plus yy is 35 squared. So this is the square root of 12 squared is 144 plus 35 squared. If you know that, congratulations. Good for you. That is 1225 144 plus 1225 is 1369. So we've got the square R of 1369. And again, if you know that without a calculator, wonderful. But the square root of 1369 is equal to 37. So our R is 37 and then we can plug that in for the cosine of theta, that's X divided by R which is 12 divided by 37. And for the sine of theta, that's Y divided by R which is 35 divided by 37. And now we can plug those in for our sine theta and cosine theta. So we have two multiplied by sine theta is 35 divided by 37. And then that's multiplied by the cosine theta which is 12 divided by 37. And if you scratch all this by hand or can do this in your head, great. But when we take two multiplied by 35 multiplied by 12, that's going to be 840 divided by and again, 37 squared, that's 1369. And that's our expression simplified. We look at our answer choices and this matches with answer choice. See, well done. We'll catch you on the next one.

Evaluate each expression without using a calculator.
sin (2 tan⁻¹ (12/5))

Key Concepts

Inverse Tangent Function (tan⁻¹ or arctan)

Double-Angle Identity for Sine

Right Triangle Trigonometry for Exact Values

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