 In this video, we provide the solution to question 22 from the practice final exam for math 1060, in which case we're given the trigonometric expression sine of two times tangent inverse of three fourths. We need to find the exact value of this thing. So we can't just throw it in a calculator and get an approximation, we need the exact value. Now, whenever you have an inverse trigonometric function like this, tangent inverse of three fourths, you should think of it as an angle, angle theta. And associated to that is a right triangle, for which we can then draw a triangle to help us process this, but also to show our work. It's a right triangle associated to theta, for which we know that tangent inverse of three fourths is equal to theta. This tells us that tangent theta is equal to three fourths. We're thinking of this as opposite over adjacent. So we get three over four, the hypotenuse by the Pythagorean equation would then be five. So this is the angle theta associated to the standard three, four, five triangle here. So what we have to compute is sine of two theta, for which this triangle here has to do with theta. It doesn't have to do with two theta. So what can we do? Well, we can apply the double angle identity to sine here and get that this is sine of two theta is the same thing as two times sine of theta times cosine of theta. So using the triangle, sine theta is gonna be opposite over hypotenuse. That's the same thing as three fifths. And cosine is adjacent over hypotenuse. That's gonna be four fifths like so. And so multiplying that together, two times three is six times four is 24, five times five is 25. And so we see that sine of two arc tangent of three fourths would equal 24 over 25.