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Right Triangle Trigonometry

Defining and Using Trigonometric Ratios


Similar Right Triangles

Suppose and are right triangles and share another congruent angle.

By the Angle-Angle Similarity Theorem, these triangles are similar. As a result, corresponding sides are proportional. In fact, the ratio between corresponding sides is constant. For example, The second proportion yields trigonometric ratios — ratios that relate the side lengths of one right triangle. Namely, the tangent of is defined as

Because proportions of all corresponding sides can be written, there are trigonometric ratios that relate all pairs of the three sides of a triangle.

Trigonometric Ratios

A trigonometric ratio relates two side lengths of a right triangle. The three most notable are sine, cosine, and tangent. Consider the right triangle

As drawn, is the hypotenuse of The remaining sides can be named relative to the marked angle Because is next to it is called the adjacent side. Similarly, because lies across from it is called the opposite side.

Sine, cosine, and tangent of are defined as follows. Trigonometric ratios can be used to determine unknown side lengths in right triangles.


Determine sine, cosine, and tangent of the angle

Show Solution

To begin we need to determine the triangle's sides relative to The side across from the right angle is always the hypotenuse. The side next to is the adjacent side and the side across it is the opposite side. Thus, Now that the sides are known, we can use the definition of sine to determine its value.

Thus, We can find the values of cosine and tangent in the same way.
ratio definition value


Sine and Cosine of Complementary Angles

By the Interior Angles Theorem, the sum of the interior angle measures of a triangle is For a right triangle, since one angle measures the other two angles are acute.

It follows that the sum of the measures of and is Therefore, they are complementary angles. The sine and cosine ratios of complementary angles have a special relationship. To explore this, the three sides in the triangle will be labeled and

The definitions of sine and cosine can be applied as follows. It can be seen that and This is true for all pairs of complementary angles. If an acute angle is named its complementary angle can be written Thus, the relationship can be written as follows.

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