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Use Six Different Ratios of a Right Triangle

Each of the three sides of a right triangle — hypotenuse, opposite, and adjacent — has a respective length or measure. And those three lengths or measures form six different ratios. Check out the following figure, which has sides of lengths 3, 4, and 5.

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

The six different ratios that you can form with the numbers 3, 4, and 5 are

image1.png

These six fractions are all that you can make by using the three lengths of the sides. The ratios are special because they represent all the possible output values of the trig functions for the acute angles in that triangle — in short, you can figure out the value of an unknown angle in a right triangle just by performing a special type of trig function on one of these ratios.

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