Trigonometry is the branch of mathematics concerned with triangles and the relationships between their angles and sides. In fact, in any given right triangle, a function known as “sine,” abbreviated sin, relates the ratio between the opposite side of an angle and the hypotenuse. Using this knowledge of the ratio of the opposite side and the hypotenuse, you can calculate the specific angle in the triangle that produced the two sides.

Determine your angle of interest. In a right triangle, you will find the following three angles: a 90 degree or right angle and two acute angles less than 90 degrees. First decide which acute angle you would like to solve for, as this will determine which side is opposite your angle of interest.

Calculate the measure of each side. Normally you will have at least two sides. You can solve for any missing side by using the Pythagorean Theorem, which states the sum of each leg-squared equals the hypotenuse-squared. For instance, if you had an adjacent of 3 and a hypotenuse of 5, then you would take the square root of 5^2 – 3^2 = sqrt(25 – 9) = sqrt(16) = 4. So your opposite side would be 4.

Divide the measure of the opposite side of your angle by the measure of your hypotenuse. For example, if your opposite side is 4 and your hypotenuse is 5, then divide 4 by 5, giving you 0.8.

Make sure the computed ratio is present on your calculator and hit the sin^-1 key. This “inverse sine” function takes a known ratio and returns the angle that produced that ratio. For example, sin^-1(0.8) = 53.130 degrees. On some calculators, you may have to hit the sin^-1 key first, type in your ratio and then press enter. Either way, once you have your angle, you can figure out the remaining angle by subtracting your result from 90. In the case of a 3-4-5 triangle, you would have 36.870, 53.130 and 90 as your three angles.

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Sky Smith has been writing on psychology, electronics, health and fitness since 2002 for various online publications. He graduated from the University of Florida with honors in 2005, earning a Bachelor of Science in psychology and statistics with a minor in math.

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