Solving the angles of a right triangle can be easy when two of them are already known. Since all of them add to 180 degrees, it’s a matter of subtracting the two known angles from 180 to find the third. When the only known angle is the 90 degree angle, but the lengths of the sides are known, it’s still possible to find the angles. In these cases, it’s necessary to use the trig ratios to find the remaining angles.
Anatomy of a Right Triangle
Any right triangle is going to have three sides and three angles. One of the angles is going to be 90 degrees and the remaining two angles will add up to 90 degrees, with 180 degrees in total. With right triangles, the side directly across from the right angle is the hypotenuse. For either of the other angles, there is an opposite side and an adjacent side.
The Six Trig Ratios
With a right triangle, there are six possible ratios to use depending on which angle is being solved. Each ratio involves a trig function and two of the sides to determine what the angle is. Only one ratio needs to be used for each triangle, as that will mean two angles are known and it’s easy to find the third. The six trig ratios are:
Sin = opposite/hypotenuse
Cos = adjacent/hypotenuse
Tan = opposite/adjacent
Csc = 1/sin = hypotenuse/opposite
Sec = 1/cos = hypotenuse/adjacent
Cot = 1/tan = adjacent/opposite
To Solve Using the Trig Ratios
To solve for an angle, determine which angle is going to be used and choose the correct ratio. Then, simply plug in the length of the sides based on the description relative to the angle and solve the ratio. Use a calculator to solve the trig function and determine the value of the angle. Since one angle is 90 degrees, once a second angle is solved, subtract the second angle from 90 degrees to find the third angle of the triangle.
Knowing Which Ratio to Use
If the problem is simply to find all of the angles of a triangle, it’s possible to use just one of the first three ratios, sin, cos or tan. If the problem is to use a certain ratio, it’s helpful to know how to solve for that ratio. For csc, sec and cot, it’s simply placing the side in the opposite position of the ratio. For example, sec is hypotenuse/adjacent, while sin is adjacent/hypotenuse.
The trig ratios can be incredibly helpful in determining the angles of a right triangle. One of the ways to remember the first three ratios is to use SOHCAHTOA, which stands for the trig function and the two parts of the ratio for all three. Sin is Opposite over Hypotenuse, Cos is Adjacent over Hypotenuse, and Tan is Opposite over Adjacent. The second set of three can be easily remembered by knowing they’re the inverse of the original three ratios. Knowing what each of the above letters stands for and how to find the inverse ratios can make it possible to solve all three angles of a right triangle where the sides are known but the only known angle is the right angle.