How to Calculate Angles in Trigonometry
- 1). Determine if the triangle you are looking at is a right triangle (this information is usually given in the problem). Right triangles have an angle of 90 degrees (also known as a right angle). The sides forming the right angle in this triangle are called legs, and the side opposite the right angle is called the hypotenuse. If you are looking at a right triangle, continue. Solving angles in other triangles is beyond the scope of this article.
- 2). Use the following trigonometric identities based on the information provided:
sin (angle) = length of the leg opposite the angle divided by the length of the hypotenuse
cos (angle) = length of the leg adjacent to the angle divided by the length of the hypotenuse
tan (angle) = length of the leg opposite the angle divided by the length of the leg adjacent to the angle
For example: If you are told that the side opposite the unknown angle is 5 cm long, and the hypotenuse is 10 cm long; then sin (angle) would be equal to 5/10, which is equal to 0.5 or 1/2.
- 3). Set your scientific calculator to the units you want. These may be degrees, radians or grads. If you are unsure how to do this, look at the instructions that came with the calculator. Having the calculator in the wrong mode will give you an incorrect answer.
- 4). Type the answer from Step 2 into your calculator. In our example we obtained sin (angle) = 0.5, so we would type 0.5 into the calculator.
- 5). Hit the corresponding inverse trigonometric key to solve the problem. The inverse trigonometric keys are the keys labeled "sin^-1," "cos^-1" and "tan^-1." In our example we found out that sin (angle) = 0.5 . We now need to find sin^-1 of 0.5 to determine the value of our unknown angle. After plugging 0.5 into the calculator and hitting "sin^-1" we discover that our unknown angle is 30 degrees.