How to Convert Tangents to Degrees

By Jennifer Ratliff
Using your calculator, figure out how to convert tangents to degrees.

The tangent has many practical uses in Physics, Calculus and Engineering, not to mention its importance in understanding fundamental concepts of geometry. The tangent is defined as rise/run or y/x. In right triangles, it is often denoted by the measure of opposite/adjacent. Here's how to convert tangents to degrees.

Figure out the tangent if you don't already know it. With a right triangle, the tangent of an unknown angle in radians is going to be opposite/adjacent. For example, if you have a triangle with hypotenuse=2, opposite=1, adjacent=1.732, your equation would be 1/1.732=.577. So your tangent would be .577 radians.

Use the formula: Tan [Math symbol theta]=opposite/adjacent (rad). Using our example, Tan [Math symbol theta]=.577 rad. In order to determine how many degrees this would be, we need to isolate the angle measurement, [theta].

Divide each side by Tangent, which will isolate your angle measurement. You should then have this equation: [Math symbol theta]=[(opp/adj) rad]/ tan. This is equivilent to saying (opp/adj)_(1/tan). But because 1/tan is simply the reciprocal of tan, we can multiply (opp/adj) and tan^-1, also known as the arctangent. On a calculator, look for the atan or tan^-1 button. Using .577 as the tan in radians, we would have .577_tan^-1=29.985 degrees.

Round to the appropriate decimal place. This usually involves rounding to the nearest degree or tenth of a degree. For example, if you have 29.985 degrees, you would round up to 30 degrees.

Double check your answer. You can check the answer by taking the tangent of your final result in degrees. This is written as follows: tan [final measure in degrees] =?. Using 30 degrees as an example, you have tan 30=?, which results in .577, which is your original number in radians.

Tip

Be certain that you are using the degree mode in your calculator. There should be a sign showing when you are using radians, RAD, but to test the calculator, computer the tangent of 45 degrees. Type Tan 45. If the answer is 1, you are in the correct mode. If you get 1.619, you're in radian mode.

One good way to remember how to figure out trig functions is with SOHCAHTOA, with H=hypotenuse, A=adjacent, O=opposite, S=sine, C=cosine, T=tangent. Remember this saying: Some Of Harry's Children Are Hungry, The Others Aren't.

Warning

Do not confuse the arctangent/inverse tangent with the cotangent. They are not the same things. The cotangent is adj/opposite, and the arctangent is the inverse of the tangent, or tan^-1. Arctangent is used to find the angle of a right triangle when the ratio "adj/opposite" is provided.

About the Author

Jennifer Ratliff has been writing Web content since 2004. Her previous clients include Elika Associates, Sam's Club and Microcap Review. She has also had work published through Odyssey and Seven Seas Magazine. Ratliff has taken courses through Ohio State University and Prestonsburg Community College. Her articles specialize in the natural sciences and mathematics.