If 𝑥 is the degree measure of an angle between zero degrees and 90 degrees and sin of 𝑥 degrees equals five over 13, what is the value of tangent of 𝑥 degrees?
Because we know that the value of 𝑥 is between zero and 90 degrees and we’re given a sine relationship. We can put this information into a right triangle. We know that sin of 𝜃 equals the opposite side length over the hypotenuse. If this is our 𝑥 degrees, we have the opposite, the hypotenuse, and we call the third side the adjacent side. Since sin of 𝑥 equals five over 13, we know the opposite side length is five and the hypotenuse side we label as 13. Once you do that, we should immediately think special right triangle, which is a triangle in ratio 5, 12, 13. We know that inside a right triangle we have a hypotenuse of 13 and one side of five. And so by this special right triangle ratio, we fill in our third side for 12.
But now, we’re trying to find the value of tangent of 𝑥. We know that tangent of 𝜃 equals the opposite over the adjacent. Tangent of 𝑥 degrees has an opposite side length of five and adjacent side length of 12. This means that the tangent of 𝑥 degrees is equal to five over 12. But what if you didn’t remember this special right triangle relationship? Let’s go back to the stage when we only knew two of the three sides in this right triangle.
In this case, you would solve with the Pythagorean theorem. 𝑎 squared plus 𝑏 squared equals 𝑐 squared. We would take five squared, the adjacent side squared. Add them together and they would be equal to 13 squared, the hypotenuse squared. Five squared is 25. 13 squared is 169. We have 25 plus 𝑎 squared equals 169. If we subtract 25 from both sides, we find that 𝑎 squared equals 144. To find 𝑎, we take the square root of both sides. And by Pythagorean theorem, we can say that 𝑎 equals 12. The square root of 144 equals positive or negative 12. But we can’t have negative distance. So we just leave this as positive 12 which confirms this 5, 12, 13 relationship. And the tangent of 𝑥 degrees must be equal to five over 12.