Given the right triangle below, evaluate tanθ\tan\thetatanθ.

tan ⁡ θ = 3 4 \tan\theta=\frac34 tan θ = 4 3

Given the right triangle below, evaluate tan θ \tan\theta tan θ .

Hey, everyone. So let's try solving this problem. Here it says, given the right triangle below, evaluate the tangent of theta, and we've got our angle theta right there. Now to solve this problem, we need to recall SOHCAHTOA. And SOHCAHTOA is this memory tool that reminds us of the main trig functions that you're going to see, which is sine, cosine, and tangent. Now in this problem, we are dealing with the tangent of theta, so what I'm going to do is focus on this relationship right here. This is that tangent is opposite over adjacent. So we can write that our tangent the the tangent of our angle theta, this angle, is equal to the opposite side of the triangle divided by the adjacent side of the triangle. Now looking at this angle, the opposite side is going to be the side that opposes the angle, which is 12. And the adjacent side is going to be this side, which is right next to the angle, which is 16. Also, something to remember is that even though this side of the triangle is also close to the angle, the longest side of the triangle is always called the hypotenuse. So we know that the adjacent side is going to be shorter than the hypotenuse, so that's how we knew this was the adjacent side. Now we have that our tangent of theta is equal to 12 over 16, but it looks like this fraction can actually reduce because 12 is the same thing as 3 times 4. Right? And then this is all equal to our tangent of theta. And then 16 is the same thing as 4 times 4. Notice how we can actually cancel a 4 in this equation, meaning that our tangent of theta is going to equal 3 4ths because we can see that that's what things reduce to when we cancel the 4ths. So as you can see, this right here is going to be the simplified relationship for the tangent of theta, and that is the answer to this problem. So hope you found this video helpful. Thanks for watching.