Chapter 6 – Trigonometric Functions: Right Triangle Approach Trigonometric Functions of Angles

Let POQ be a right triangle with an acute angle  where  is in standard position. The point P is on the terminal side of  Trigonometric Functions of Angles P(x, y) y x r Q O x y

Definition of Trigonometric Functions Let  be an angle in standard position and let P(x, y) be a point on the terminal side. If is the distance from the origin to the point P(x, y), then Trigonometric Functions of Angles

Quadrantal Angles Quadrantal Angles are angles that are coterminal with the coordinate axis Trigonometric Functions of Angles

Remember: Signs of the Trig Functions QuadPositive Functions Negative Functions I All None II sin, csccos, sec, tan, cot III tan, cotsin, csc, cos, sec IV cos, secsin, csc, tan, cot Trigonometric Function of Real Numbers

Examples – pg. 460 Find the quadrant in which  lies from the information given Trigonometric Functions of Angles

Reference Number Let  be an angle in standard position. The reference angle   associated with  is the acute angle formed by the terminal side of  and the x- axis The Unit Circle

Evaluating Trig Functions for Any Angle To find the values of the trigonometric functions for any angle , we carry out use the following steps: 1. Find the reference angle   associated with the angle . 2. Determine the sign of the trigonometric function of  by noting the quadrant in which  lies. 3. The value of the trigonometric function of  is the same, except possibly for sign, as the value of the trigonometric function of   The Unit Circle

Examples – pg. 459 Find the reference angle for the given angle Trigonometric Functions of Angles

Examples – pg. 459 Find the exact value of the trigonometric function Trigonometric Functions of Angles

Fundamental Identities Reciprocal Identities Pythagorean Identities Trigonometric Function of Real Numbers

Examples – pg. 460 Write the first trigonometric function in terms of the second for  in the given quadrant Trigonometric Functions of Angles

Examples – pg. 460 Find the given values of the trigonometric functions of  from the information given Trigonometric Functions of Angles

Area of a Triangle The area A of a triangle with sides of length a and b with included angle  is Trigonometric Functions of Angles

Examples – pg Trigonometric Functions of Angles

Examples – pg Trigonometric Functions of Angles