Appendix D Trigonometry Angles can be measured in degrees or radians The angle given by a complete revolution contains 360o or 2 radians 360o 2 rad o 180 1 rad 1o rad 180 Example 1 a Convert 9o to radians b Convert 5 to degrees 12 The standard position of an angle occurs when we place its vertex at the origin of a coordinate system and its initial side on the positive x axis A positive angle is obtained by rotating the initial side counterclockwise until it coincides with the terminal side Negative angles are obtained by clockwise rotation 1 The trigonometric functions adj hyp adj cot opp hyp csc opp opp hyp opp tan adj hyp sec adj cos sin Special triangles 2 3 1 sin sin sin 4 2 6 2 3 2 2 3 1 cos cos cos 4 2 6 2 3 2 1 tan 1 tan sin 3 4 6 3 3 2 Signs of the trigonometric functions Example 2 a If sin 3 5 0 2 find the remaining trigonometric rations b If cos 31 3 2 find the remaining trigonometric rations 3 Trigonometric Identities csc 1 sin 1 1 cot cos tan 2 2 sin cos 1 sec tan sin cos tan2 1 sec2 cot2 1 csc2 sin sin cos cos sin 2 sin cos 2 cos tan tan cot cot sin x y sin x cos y cos x sin y sin x y sin x cos y cos x sin y cos x y cos x cos y sin x sin y cos x y cos x cos y sin x sin y tan x tan y tan x y 1 tan x tan y tan x tan y tan x y 1 tan x tan y sin 2x 2 sin x cos x Example 4 the expression a cos x y cos 2x cos2 x sin2 x 1 2 sin2 x 2 cos2 x 1 1 cos 2x cos2 x 2 1 cos 2x sin2 x 2 1 sin x cos y sin x y sin x y 2 1 cos x cos y cos x y cos x y 2 1 sin x sin y cos x y cos x y 2 5 1 If sin x and sec y where x and y lie between 0 and 2 evaluate 3 4 4 b sin 2y Example 5 Find all values of x in the interval 0 2 that satisfy the equation a 2 cos x 1 0 b 2 cos x sin 2x 0 5 Graphs of the trigonometric functions y sin x y cos x y tan x y cot x 6 y csc x y sec x Example 6 Sketch the graph of f x 1 sin x 7
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