How To Compute Sine, Cosine and Tangent

Sine, cosine and tangent are values that are derived from the ratios between the sides of triangles. These values are often encountered in the branch of mathematics called trigonometry.

Here's how you compute for the sine, cosine and tangent. These are all based on right-angle triangles. Before anything else, it's best if we define the terms to be used.

These functions are computed relative to any of the two angles in a right triangle except for the 90o angle that makes the right triangle right. The opposite side is the leg of the triangle that is directly opposite to the angle you are taking the function (sine, cosine or tangent) of. The adjacent side is the leg that is directly connected to the angle, but is not the hypotenuse, which is is the longest side of the triangle (usually also the leg directly opposite the right angle).


Use the trigonometric mnemonic SOH CAH TOA. This is a nifty way of remembering how each of the trigonometric functions could be computed based on the sides of the triangle.

Sine= opposite / hypotenuse

For the sine function, all you have to do is to divide the length of the opposite side opposite the angle by the measurement of the hypotenuse. The hypotenuse is the triangle's longest side.

Cosine = adjacent / hypotenuse

For the cosine function, divide the length of the adjacent side relative to the angle with the length of the hypotenuse.

Tangent = opposite / adjacent

For the tangent function, divide the length of the opposite side relative to the angle by the length of the adjacent side.

Always remember that the values are derived from the angles of the triangle. When written down, for instance it should look like this: sin 90o or sin 90 degrees.


You may also use a calculator to solve for trigonometric functions. This becomes very helpful when solving physics problems where most values come in vectors.

Some calculators would require you to input the angle value first before pressing the desired trigonometric function.

There are some that would require the exact opposite - you would have to press the trigonometric function first before providing the angle.

Just be aware of your calculator's quirks. Read the manual. If you don't want to read the manual, just observe how the calculator reacts to the values put into it. If you are doing it wrong, it would usually alert you with an "error" message. Or you might have to compute for the results manually first, so you know how to use your calculator properly.

Trigonometric functions have no units. Since you are dividing two lengths, the units effectively cancel each other out. What's left is a ratio with no units. These functions are often useful when you are computing for missing pieces of the puzzle. For instance, you might have a triangle where only two of the side lengths are given. If you know the angles, and you know the computations, you can use these as variables in your SOH CAH TOA equations.

Whether you prefer using the long hand method or a calculator, mathematics would always require plenty of practice. Pick up sample problems from your algebra book, or search online for exercises and word problems involving trigonometric functions.


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