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The Sine Function: Opposite over Hypotenuse

When you’re using right triangles (triangles with right angles in them) to define trig functions, the trig function sine, abbreviated sin, has input values that are angle measures and output values that you obtain from the ratio

image0.png

The below figure shows two different acute angles, and each has a different value for the function sine.

A right triangle with sides of lengths 3, 4, and 5.
A right triangle with sides of lengths 3, 4, and 5.

The two values are

image2.png

The sine is always the measure of the opposite side of the acute angle divided by the measure of the hypotenuse. Because the hypotenuse is always the longest side, the number on the bottom of the ratio will always be larger than that on the top. For this reason, the output of the sine function of any acute angle will always be a proper fraction — it’ll never be greater than 1. And it’s only equal to 1 when you’re working with circles, not triangles.

Even if you don’t know both lengths required for the sine function, you can calculate the sine if you know any two of the three lengths of a triangle’s sides. Thank goodness for Pythagoras.

For example, to find the sine of angle a in a right triangle whose hypotenuse is 10 inches long and adjacent side is 8 inches long, follow these steps:

  1. Find the length of the side opposite a.

    Use the Pythagorean theorem, letting a be 8 and c be 10. When you input the numbers and solve for b, you get

    image3.png

    So the opposite side is 6 inches long.

  2. Use the ratio for sine, opposite over hypotenuse.

    image4.png
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