13 3 Trigonometric Functions of General Angles Objectives

13. 3 Trigonometric Functions of General Angles Objectives: 1. Find values of trig functions for general angles 2. Use reference angles to find values of trig functions

Trig Functions, θ in Standard Position You already know how to find the trig values of a right triangle with respect to either of the acute angles. In this lesson you will learn how to find the values for any angle. If you are given an ordered pair, create a right angle, find the hypotenuse, the use Soh Cah Toa.

Trigonometric Ratios of Angles of any measure. • Let be an angle in standard position and let P(x, y) be a point on the terminal side of . Using the Pythagorean Theorem, the distance r from the origin to P is given by The trigonometric functions of the angle in standard position may be defined as follows.

Example Find the exact vales of the six trig functions of θ if the terminal side of θ in standard position contains the point (8, -15) x=8, y=-15, r=17 (8, -15)

Quadrantal Angles Quadrantal angles have terminal sides that lie on one of the axes. Degrees – 0, 90, 180, 270 Radians – Two of the trig values are undefined for quadrantal angles because it involves division by 0.

Find the values of the six trig. functions for an angle in standard position that measures 180° When = 180°, x = -1 and y = 0 r=1 sin 180° = csc 180°= cos 180°= sec 180°= tan 180°= cot 180°= -1, 0

Reference Angles Reference angles are used to find the values of trig functions greater than 90°. If θ is a nonquadrantal angles in standard position, its reference angle, θ’ is defined as the acute angle formed by the terminal side of θ and the x-axis

Reference Angle Rules Quadrant I θ’=θ Quadrant II θ’=180 -θ or π-θ θ’ θ θ θ Quadrant III θ’=θ-180 or θ-π Quadrant IV θ‘=360 -θ or 2π-θ θ’ θ θ'

Example Sketch each angle. Find each reference angle. 1. 236° 236 -180=56 2. Coterminal angle:

Trig Values for Special Angles θ sin θ cos θ tanθ 30° 45° 60° csc θ sec θ cot θ 2 1 1 2

Signs of trig functions Quadrant Function I II IV sinθ or csc θ + + - - cosθ or sec θ + - - + tan θ or cot θ + -

Finding exact trig values Find the reference angle for each angle. Keep in mind the signs are based on the quadrant. 1. sin 135° second quadrant – sin is positive reference angle = 45° at 45°, the value of sin = 2. cot 1 st quad – cot is positive is the reference angle at (60°), cot=

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