Hey, everyone. You may come across this specific type of problem in which you're given certain information about theta and asked to find a function of 2 theta. Now this is really similar to other problem types that we've seen before, but now just for our double angle identities. So let's go ahead and jump in here. We're gonna use the same exact steps that we saw with our sum and difference identities when doing a really similar problem type. So let's look at this problem. We are given the cosine of theta is equal to 4/5 and that theta is in between 0 and π/2, and we want to find the sine of 2θ. Now, remembering our steps here in evaluating trig functions when we're given different conditions, our step number 1 is to expand our identity out and identify any of our unknown trig values. Now here we're asked to find the sine of 2θ and when we find the sine of 2θ we're going to look at this identity here. The sine of 2θ is equal to 2 times the sine of theta times the cosine of theta. Now we want to identify our unknown trig values, and in our problem statement, we're told that the cosine of theta is equal to 4/5. So we already know that and all that's left to find is the sine of theta. So let's move on to our next step and figure out how to get there. Step 1 is done. In step 2, we're told that from our given info, we want to sketch and label our triangles in the proper quadrant, remembering to pay close attention to the sign. Here we're told that the cosine of theta is equal to 4/5 and that theta is in that first quadrant. So I'm going to go ahead and draw my right triangle here and label my angle theta on that inside. Now since the cosine of this angle theta is equal to 4/5 that tells me that this adjacent side is 4 and my hypotenuse is 5 and we have completed step number 2. Now, from step number 3, we want to find any missing side lengths using the Pythagorean theorem. Now here, I don't even need to use the Pythagorean theorem because I see that this is a 3, 4, 5 triangle. So my missing side length is simply 3 and step number 3 is done. Now moving on to step number 4, we want to solve for any unknown trig values that we identified in step number 1. Now in step 1, we needed to find the sine of beta. So let's go ahead and do that here. Now the sine of beta is gonna be equal to that opposite side that we just found, 3 over that hypotenuse 5, and that's my only unknown trig value. So we've completed step number 4. Now with that unknown trig value, we can move on to step number 5 and plug in all of our values and simplify. So here I'm taking 2, I'm multiplying it by that sine of theta that I just found in step 4, 3/5, and then multiplying it by that cosine that I was given in my problem, 4/5. Now all that’s left to do is simplify doing some algebra here. Now multiplying 2 times 3 times 4 will give me 24. Then in my denominator, I just have 5 times 5, which is 25. So that gives me my final answer here. Step 5 is done and I have successfully solved this problem with my final answer 24/25. Thanks for watching, and I'll see you in the next one.