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Section 4.4 Trigonometric Functions of Any Angle Overview • In this section we will find trigonometric values for any angle. • Doing so requires to consider rotations for circles centered at the origin but not with a radius of 1. • We will use right triangle trigonometry to write the appropriate ratios. Definitions y sin r x cos r y tan x r csc y r sec x x cot y Examples 1. The point (-9, 12) is on the terminal side of an angle θ. Find the exact value of each of the six trigonometric functions of θ. 2. The point (-5, -4) is on the terminal side of an angle θ. Find the exact value of each of the six trigonometric functions of θ. Trig Values And Their Signs • Recall the signs of the coordinates of x and y and each of the four quadrants: • The six trigonometric values follow the same rules and patterns. Students All Take Calculus Examples 1. Given that cos θ = -4/5 and θ is in Quadrant II, find the remaining five trigonometric values of θ. 2. Given that tan θ = 12/5 and cos θ < 0, find the remaining five trigonometric values of θ. Reference Angles • • 1. 2. 3. 4. A reference angle is a positive acute angle formed by the terminal side and the x-axis. To find a reference angle, first find find the coterminal angle for the given angle that is between 0º and 360º or 0 radians and 2π radians. Note the quadrant of the coterminal angle. If the coterminal angle is in Quadrant I, do nothing. The coterminal angle is your reference angle. If the coterminal angle is in Quadrant II, subtract it from 180º or π radians. If the coterminal angle is in Quadrant III, subtract 180º or π radians from the angle. If the coterminal angle is in Quadrant IV, subtract it from 360º or 2π radians. Examples • Find reference angles for each of the following: 112 272 7 6 23 4