Hey everyone. In this problem, we're asked to evaluate the expression the sine of the inverse tangent of 2 thirds. Now, looking at this composite trig function, my inside function is an inverse, and that value is not something I recognize as being on the unit circle, so I know that I need to solve this by using a right triangle. So let's go ahead and jump into our steps here. Now, step number 1 is to use our interval to identify our quadrant. Now we're, of course, working with our inside function first here. So our inside function is the inverse tangent. And the inverse tangent has an angle interval from negative pi over 2 to positive pi over 2. So I know that my right triangle either needs to be in quadrant 1 or quadrant 2 to match that interval. Now here we want to consider the sign of our argument. Now our argument here is a positive value of 2 thirds. And I know that within this interval, my tangent values can only be positive in quadrant 1, so that's where my triangle is going to go. Now we've completed step number 1. We can move on to step number 2. And now we want to go ahead and actually draw that right triangle in quadrant 1 with that side length down against the x-axis. And then I'm going to label my angle theta as that inside angle right there. Now looking at my argument, I have 2 thirds, and my inside function tells me that the tangent of that angle that I just labeled has to be equal to that 2 thirds argument. So based on SOHCAHTOA, I know that my opposite side has to be 2 and my adjacent side has to be 3. Now, we've completed step number 2. Moving on to step number 3, we want to use the Pythagorean theorem in order to find that missing third side. Now here our missing side length is our hypotenuse. So setting up our Pythagorean theorem here, a2+b2=c2. I'm solving for c here because my missing side is the hypotenuse. So here I take 22+32, those two side lengths that I already know. That's equal to c squared. Now 22 is 4, and 32 is 9. That's equal to c squared. Adding those two values together, I get that 13 is equal to c squared. Now my final step here in getting this missing side length is taking the square root of both sides. Now root 13 cannot be simplified anymore, so that's just what my side length is. Root 13 is equal to my hypotenuse, that missing side length. So I can go ahead and label that up here on my triangle. And we have completed step number 3. Now moving on to our final step, we are going to use our right triangle in order to evaluate that outside function. Now here, my outside function is the sine. So I am tasked with finding the sine of my angle theta, which I know I need my opposite side and my hypotenuse based on SOHCAHTOA. So I have all the information I need here. Looking at my triangle, I'm going to take that opposite side of 2 and divide it by my hypotenuse, which is the square root of 13. Now this is technically a correct answer. But whenever we have a radical in the denominator, we want to go ahead and rationalize that denominator, which we can do here by multiplying this by root 13 over root 13. So that gets our radical, out of that denominator, leaving me with 2 root 13 on the top and just 13 on the bottom. So my final answer here is that the sine of the inverse tangent of 2 thirds is equal to 2 root 13 over 13. Thanks for watching, and let me know if you have questions.