Find the main trigonometric ratios of angle 𝐴, given that 𝐴𝐵𝐶 is a right triangle at 𝐵, where the ratio between 𝐴𝐵 and 𝐴𝐶 is four to five.
The main trigonometric ratios are the sine, cosine, and tangent ratios. In any right triangle, the sin of angle 𝜃 is equal to the opposite over the hypotenuse. The cos of angle 𝜃 is the adjacent over the hypotenuse. And the tan of angle 𝜃 is the opposite over the adjacent. One way of recalling these three ratios is using the acronym SOHCAHTOA.
In this question, we have a right triangle 𝐴𝐵𝐶, where the right angle is at point 𝐵. We are told that the ratio between side 𝐴𝐵 and 𝐴𝐶 is four to five. Since 𝐴𝐶 is the hypotenuse of our right triangle, we can use our knowledge of Pythagorean triples to conclude that the three sides will be in the ratio three to four to five. The ratio of 𝐵𝐶 to 𝐴𝐵 to 𝐴𝐶 is three to four to five.
The angle we are interested in this question is angle 𝐴. This means that side 𝐵𝐶 is the opposite, as it is opposite angle 𝐴. 𝐴𝐵 is the adjacent, as it is next to angle 𝐴 and the right angle. And finally, 𝐴𝐶 is the hypotenuse, as it is the longest side of our right triangle and is opposite the right angle. As sin 𝜃 is equal to the opposite over the hypotenuse, the sin of angle 𝐴 is equal to three-fifths. The cos of angle 𝐴 is equal to four-fifths. This is the adjacent over the hypotenuse. And finally, the tan of angle 𝐴 is equal to three-quarters. This is the opposite over the adjacent. These are the three main trigonometric ratios of angle 𝐴.