Video Transcript
Find the main trigonometric ratios of angle ๐ถ given that ๐ด๐ต๐ถ is a right-angled triangle at ๐ต, where 20 times the tangent of ๐ด 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 ๐ต. The other piece of information that we were given is that 20 times tan ๐ด equals 21. We remember from our trig ratios that the tangent of angle ๐ด 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 ๐ด equals 21, and put it in the format of opposite over adjacent. To do that, weโll need to divide both sides of this equation by 20. 20 divided by 20 cancels out leaving us with tan ๐ด equals 21 over 20.
Weโll use this information to label our triangle. Our starting point is angle ๐ด. 21 would be the side length opposite angle ๐ด, and 20 would be the adjacent side length to angle ๐ด. We were looking for the main trig ratios of angle ๐ถ. By main trig ratios, we mean the sine of ๐ถ, the cosine of ๐ถ, and the tangent of ๐ถ. The sine of ๐ถ would be the opposite side length over the hypotenuse, the cosine of ๐ถ would be the adjacent side length over the hypotenuse, and the tangent of ๐ถ 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 ๐ถ and the adjacent side length to angle ๐ถ. 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 ๐ด๐ถ? Because this is a right triangle, we can use the Pythagorean theorem. We can say 20 squared plus 21 squared equals ๐ถ squared. 400 plus 441 equals ๐ถ squared; 841 equals ๐ถ squared. We take the square root of both sides, and we find that ๐ถ โ our missing side length, our hypotenuse โ 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 ๐ถ, the opposite side length is 20 and the hypotenuse equals 29. Our last ratio to complete is the cosine of ๐ถ. The adjacent side length is 21 and again the hypotenuse is 29.
The main trig ratios for angle ๐ถ are: sin ๐ถ equals 20 over 29, cos ๐ถ equals 21 over 29, and tangent ๐ถ equals 20 over 21.
