In a right triangle, the hypotenuse is always the longest side of the triangle and lies across from the right angle. The triangle's other two sides are called legs. If one of the acute angles in the triangle is labeled, the legs can be discussed relative to that angle. If is labeled as shown, the side that forms the angle is the adjacent side and the side not touching the angle is the opposite side.
The trigonometric functions only state a ratio between two sides in a right triangle, they do not give the lengths of the individual sides. However, if one acute angle and one side is known, the lengths of the other sides can be determined.
Evaluate the trigonometric functions for the angle
To begin, we should label the triangle's sides. The longest side is the hypotenuse, the side next to is the adjacent side and the side across from is the opposite side.
We can now evaluate the trigonometric functions by using the definitions and the lengths of the sides in the triangle.
For a right triangle with legs and and hypotenuse the following is true.
For the right triangle, find and
To begin, we should note that since the given triangle is a right triangle, both the trigonometric ratios and the Pythagorean Theorem might be useful tools. The given information includes one angle and one side. Since two side lengths must be known to use the Pythagorean Theorem, we'll have to use a trigonometric ratio. We can label the sides of the triangle relative to the labeled angle.
If the value of sine, cosine or tangent for a specific angle, , is known, it is possible to calculate the measure of the angle using inverse trigonometric functions. The inverse function of sine is called "inverse sine" and is written as
In the same way, inverse cosine () and inverse tangent () are the inverses of cosine and tangent, respectively.
Determine the measure of the angle
In order to solve for we must use inverse trigonometric ratios. To begin, we'll label the sides of the triangle relative to
Since the hypotenuse and the adjacent side are given, we'll use