Trigonometric functions are based on relationships present between the sides and angles of triangles. Many functions are derived from a right triangle—a triangle containing a right (90°) angle. Consider the following right triangle with sides A, B, and C, and angles α, β, and γ.

Side C, which is the longest side and the side opposite the right angle, is known as the *hypotenuse of the triangle*.

A commonly used trigonometric relationship for right triangles is the Pythagorean theorem. The *Pythagorean theorem* is an expression of the relationship between the hypotenuse and the other two sides of a right triangle:

The sum of the squares of the lengths of the two sides of a right triangle is equal to the square of the length of the hypotenuse.

Using the sides of the labeled triangle yields the following:

Suppose that sides A and B are 3 and 4 units long, respectively. The Pythagorean theorem can be used to solve for the length of side C:

Three trigonometric relationships are based on the ratios of the lengths of the sides of a right triangle. The sine (abbreviated *sin*) of an angle is defined as the ratio of the length of the side of the triangle opposite the angle to the length of the hypotenuse. Using the labeled triangle yields the following:

With *A* = 3, *B* = 4, and *C* = 5:

The cosine (abbreviated *cos*) of an angle is defined as the ratio of the length of the side of the triangle adjacent to the angle to the length of the hypotenuse. Using the labeled triangle yields the following:

With *A* = 3, *B* = 4, and *C* = 5:

The third function, the tangent (abbreviated *tan*) of an angle, is defi ned as the ratio of the length of the side of the triangle opposite the angle to that of the side adjacent to the angle. Using the labeled triangle yields the following:

With *A* = 3, *B* = 4, and *C* = 5:

Two useful trigonometric relationships are applicable to *all* triangles. The first is known as the law of sines:

The ratio between the length of any side of a triangle and the angle opposite that side is equal to the ratio between the length of any other side of the triangle and the angle opposite that side.

With respect to the labeled triangle, this may be stated as the following:

A second trigonometric relationship applicable to *all* triangles is the law of cosines:

The square of the length of any side of a triangle is equal to the sum of the squares of the lengths of ...