 In this video, we provide the solution to question number three for practice exam number four for math 1220, in which case we have to compute the slope of the tangent line of the polar curve r equals 8 sine of theta at the moment where theta equals pi thirds. So to find the slope of the tangent line, we're looking for dy over dx, which by the chain rule, this is the same thing as dy over d theta, divided by dx over d theta, for which y, we can think of it as r times sine theta. We need to take the derivative with respect to theta. And on the bottom, we're going to have r cosine theta take the derivative here. Now in this situation, r is equal to 8 sine. So I'm going to make that substitution in here right now. You can use the general formula, but we have an 8 sine squared in the numerator we have to take the derivative. And in the bottom, we have an 8 sine theta cosine theta. Now in the bottom to make the derivative a little bit easier, I'm actually going to replace the sine theta cosine theta with actually a double angle, so sine of 2 theta, like so. I should also mention that these coefficients cancel out so you have a 2 right here. So you take the derivative of this thing on the top using the chain rule, you're going to get 4 times sine theta cosine theta. And then in the denominator, you take the derivative, you're going to get 2 cosine of 2 theta, like so. And we need to evaluate this thing when theta is equal to pi thirds. I could try to simplify this more using trig identities, but now that the derivative is calculated, I just can evaluate it at pi thirds. So the things to remember about this thing, sine at pi thirds is going to be root 3 over 2. Cosine at pi thirds is going to equal one half. And so in the denominator, you're going to get 2 times cosine of 2 pi thirds. So we'll come back to that one in a second. In the top, this 4 is going to cancel with these factors of one half here. So the top is just going to become a square root of 3. In the denominator, we do have a 2 of course. So cosine, I should mention here, cosine of 2 pi thirds using reference angles in the second quadrant where 2 pi thirds lives. Cosine is negative, but it will reference to pi thirds. So this is going to give you a negative one half, like so. So in the bottom, that 2 and the negative one half will simplify to give us a negative one. And so then we end up with a slope being negative, the square root of 3, which then leads us to having as our option C as the correct answer there.