Appendix B: Trigonometric Functions

INTRODUCTION

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:

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 defined 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:

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