Sullivan Algebra and Trigonometry: Section 6.5 Unit Circle Approach; Properties of the Trig Functions Objectives of this Section Find the Exact Value of the Trigonometric Functions Using the Unit Circle Determine the Domain and Range of the Trigonometric Functions Determine the Period of the Trigonometric Functions Use Even-Odd Properties to Find the Exact Value of the Trigonometric Functions

The unit circle is a circle whose radius is 1 and whose center is at the origin. Since r = 1: becomes

(0, 1) (-1, 0) (0, -1) (1, 0) y x

(0, 1) (-1, 0) (0, -1) (1, 0) y x P = (a, b)

Let t be a real number and let P = (a, b) be the point on the unit circle that corresponds to t. The sine function associates with t the y-coordinate of P and is denoted by The cosine function associates with t the x-coordinate of P and is denoted by

(0, 1) (-1, 0) (0, -1) (1, 0) y x P = (a, b)

y x r a b

(5, 0)

(0, 1) (-1, 0) (0, -1) (1, 0) y x P = (a, b)

The domain of the sine function is the set of all real numbers. The domain of the cosine function is the set of all real numbers. The domain of the tangent function is the set of all real numbers except odd multiples of The domain of the secant function is the set of all real numbers except odd multiples of

The domain of the cotangent function is the set of all real numbers except integral multiples of The domain of the cosecant function is the set of all real numbers except integral multiples of

Let P = (a, b) be the point on the unit circle that corresponds to the angle. Then, -1 < a < 1 and -1 < b < 1. Range of the Trigonometric Functions

If there is a smallest such number p, this smallest value is called the (fundamental) period of f.

Periodic Properties

Even-Odd Properties