Given tan⁡θ=512\tan\theta=\frac{5}{12}tanθ=125 and 0 < θ\thetaθ < π2\frac{\pi}{2}2π, find cos⁡(2θ)\cos\left(2\theta\right)cos(2θ).

119 169 \frac{119}{169} 169 119

Given tan ⁡ θ = 5 12 \tan\theta=\frac{5}{12} tan θ = 12 5 and 0 < θ \theta θ < π 2 \frac{\pi}{2} 2 π , find cos ⁡ ( 2 θ ) \cos\left(2\theta\right) cos ( 2 θ ) .

In this problem we're given that the tangent of theta is equal to 5 over 12 and that theta is in between 0 and pi over 2 and then asked to find the cosine of 2 theta. Now let's go ahead and get started with our steps to evaluate trig functions given certain conditions. Now, starting with step number 1, we want to go ahead and expand our identity and then identify any unknown trig values. Now here we're asked to find the cosine of 2 theta and there are actually 3 different formulas that we could use to do that. Now, any of these 3 will work here but I'm going to go ahead and choose one that has the least amount of unknown trig values. Since I only know the tangent, the cosine and the sine are both unknown. So I'm just going to go ahead and use this second one here. The cosine of 2 theta is equal to 1 minus 2 times the sine squared of theta. Now, let's go ahead and move on to step number 2 because I know that sine is my only unknown trig value here. Now in step 2, we want to sketch our triangles labeling them and making sure to pay attention to the sine. Now here we're told that the tangent of our angle theta is 5 over 12 and that theta is in that first quadrant in between 0 and pi over 2. So labeling theta there knowing that the tangent of this angle is 5 over 12 tells me that my opposite side is 5, my adjacent side is 12. So step number 2 is done. Moving on to step 3, we wanna find any missing side lengths. Now you could go ahead and use the pythagorean theorem here but I notice that this is a special right triangle, a 5, 12, 13 triangle. So I know that that hypotenuse is equal to 13 and we have successfully found that missing side length. Now for step number 4, we want to solve for any unknown trig values that we identified in step number 1. Now we only have one unknown trig value, the sine of theta. So let's find that. The sine of theta is equal to that opposite side over that hypotenuse, so 5 over 13. Now from here, we're going to move on to step number 5 and just plug in that value. So here, plugging that value in, we end up with 1 minus 2 times that sine value, 5 over 13, that we we just found and make sure that you square that value. Now from here, we just have to simplify and do some algebra. Now I'm gonna first go ahead and square this fraction right here and that will give me 25 over 169. Now this is being multiplied by 2 and then subtracted from 1. I'm gonna go ahead and do that multiplication by 2, and this gives me 1 minus 50 over 169. Now I wanna go ahead and subtract these two fractions using a common denominator of a 169. So to do that, I need to multiply this one by 169 over 169 to give me 169 minus 50 over that common denominator of 160 9. Now from here, subtracting that 50 from a 169 on the top gives me 119, and that is still over that denominator of 100 and 69. Now here is my final answer, 119 over 169. Now this is not necessarily a pretty fraction but it is equal to the cosine of 2 theta given all this information. Thanks for watching and let me know if you have any questions.