 In this video, we will provide the solution to question number 11 for practice exam 2 for math 12-10. We are given the function g of x is sine of x, and we are allowed to assume that the derivative of g of x is cosine of x. Many of us might be able to prove that through other methods, but you can take that for granted. The derivative of sine is cosine, that's given to you. So with that knowledge, we're asked to find the equation of the line tangent to the graph at x equals pi thirds. We have to write this in slope intercept form. So when it comes to tangent lines, it's important to remember the following equation. The tangent line will be given by y minus g of a is equal to g prime at a times x minus a. The tangent line will always have that equation, probably worth putting on your no-card. In which case then we have to do specifically, we're going to take y minus, well our value here is pi thirds. So we're going to take g of pi thirds, let's sine of pi thirds, like so. Then we get g prime of pi thirds. Well the derivative as we were told is cosine. So we're going to get cosine of pi thirds, and then we times that by x minus pi thirds, like so. So then we have to evaluate either by memory or using your calculator. We're going to evaluate sine of pi thirds and cosine of pi thirds. Sine of pi thirds, pi thirds of course is 60 degrees. This will be the square root of 3 over 2. Cosine of pi thirds is one half. So this gives us an equation of the tangent line. We have to put in slope intercept form. So it's going to need to look like y equals mx plus b. Basically we need to solve for y in the setting. So the first thing I'm going to do is I'm going to distribute the one half through. This gives us one half x minus pi sixth, like so. And then we're going to add the root 3 over 2 to both sides of the equation. So it cancels here and then we have to add it over here. For which as these are two irrational numbers, leave it in exact form. Don't write it as a numerical approximation. Just leave it exact. This will give us y is equal to you can write this as either one half x or if you prefer, you can write it as x over 2. It doesn't make much of a difference to me. It's the same number either way. I'm going to leave it as one half x just to make the slope more prominent. And so then you're going to get negative pi over sixth plus root 3 over 2. For which that right there is the wider set, the combination. That would be it. If we want to write them together, we'll have to find a common denominator. So between 2 and 6, the least common multiple will be 6. So I could times the second fraction by 3 over 3. This would then give us y equals one half x plus 3 root 3 minus pi all over 6. So if you can, you want to write as a single y intercept. That's perfectly fine. Now, I want to mention the last step is not absolutely necessary. Writing it together as a common fraction, although is legitimate. It's not necessary for one like this. I would encourage you, though, not to write a numerical approximation. The instructions for this exam do say to leave answers exact unless instructed to round, which no instructions given on this question.