Find the main trigonometric ratios of angle 𝐶 given that 𝐴𝐵𝐶 is a right-angled triangle at 𝐵, where 20 times the tangent of 𝐴 equals 21.
But I think we should do here is go ahead a sketch out a triangle to help us visualize all the ratios. Our sketch is based on the fact that our question told us this triangle is right angle with angle 𝐵. The other piece of information that we were given is that 20 times tan 𝐴 equals 21. We remember from our trig ratios that the tangent of angle 𝐴 would be the opposite side length over the adjacent side length. We wanna try and take the information we were given here, 20 times tangent 𝐴 equals 21, and put it in the format of opposite over adjacent. To do that, we’ll need to divide both sides of this equation by 20. 20 divided by 20 cancels out leaving us with tan 𝐴 equals 21 over 20.
We’ll use this information to label our triangle. Our starting point is angle 𝐴. 21 would be the side length opposite angle 𝐴, and 20 would be the adjacent side length to angle 𝐴. We were looking for the main trig ratios of angle 𝐶. By main trig ratios, we mean the sine of 𝐶, the cosine of 𝐶, and the tangent of 𝐶. The sine of 𝐶 would be the opposite side length over the hypotenuse, the cosine of 𝐶 would be the adjacent side length over the hypotenuse, and the tangent of 𝐶 would be the opposite side length over the adjacent side length.
Currently, we only know two side lengths. We know the opposite side length to angle 𝐶 and the adjacent side length to angle 𝐶. Since we know the opposite side length and the adjacent side length, we can start by filling in the tangent. The opposite side length is 20, and the adjacent side length is 21. But how we know the side length of 𝐴𝐶? Because this is a right triangle, we can use the Pythagorean theorem. We can say 20 squared plus 21 squared equals 𝐶 squared. 400 plus 441 equals 𝐶 squared; 841 equals 𝐶 squared. We take the square root of both sides, and we find that 𝐶 — our missing side length, our hypotenuse — is 29.
We can label our triangle with 29 for the hypotenuse, and now we have enough information to fill in the other two ratios. Starting with sine of 𝐶, the opposite side length is 20 and the hypotenuse equals 29. Our last ratio to complete is the cosine of 𝐶. The adjacent side length is 21 and again the hypotenuse is 29.
The main trig ratios for angle 𝐶 are: sin 𝐶 equals 20 over 29, cos 𝐶 equals 21 over 29, and tangent 𝐶 equals 20 over 21.