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Properties of the Trigonometric Functions Domain and Range • Remember: • sin y 1 csc y 0 y cos x 1 sec x 0 x y tan x 0 x x cot y 0 y Domain and Range • The domain of the sine function is all real numbers. The range of the sine function is [-1, 1] • The domain of the cosine function is all real numbers. The range of the cosine function is [-1, 1] Domain and Range • The domain of the tangent function is the set of all real numbers, except odd multiples of p/2. The range is all real numbers. • The domain of the secant function is the set of all real numbers, except odd multiples of p/2. The range is (-∞, 1] u [1, ∞). Domain and Range • The domain of the cotangent function is the set of all real numbers except integral multiples of p. The range is all real numbers. • The domain of the cosecant function is the set of all real numbers except integral multiples of p. The range is (-∞, 1] u [1, ∞) Periodic Functions • Definition: • A function f is called periodic if there is a positive number p such that, whenever θ is in the domain of f, so is θ + p, and • f(θ + p) = f(θ) Periodic Properties sin( 2p k ) sin csc( 2p k ) csc cos( 2p k ) cos sec( 2p k ) sec tan( p k ) tan cot( p k ) cot Periodic Functions • If sin θ = 0.3, find the value of sin θ + • sin (θ + 2p) + sin (θ + 4p) • If tan θ = 3, find the value of tan θ + • tan (θ + p) + tan (θ + 2p) Signs of the Trigonometric Functions • Table 5 p. 403 • Remember the mnemonic (All – Quad I; Scientists – Quad II; Take – Quad III; Calculus – Quad IV) Finding the Quadrant in Which an Angle Lies • If sin < 0 and cos < 0, name the quadrant in which the angle lies. • If sin < 0 and tan < 0, name the quadrant in which the angle lies. Fundamental Identities • Reciprocal Identities: 1 csc sin 1 sec cos Quotient Identities: sin tan cos cos cot sin 1 cot tan Fundamental Identities • Pythagorean Identities: sin cos 1 2 2 tan 1 sec 2 2 cot 1 csc 2 2 Finding Exact Values of A Trig Expression 5 2 5 Given sin and cos 5 5 Find the other four trig functions using identities and/or unit circle Find the Exact Value of Trig Functions • Find the exact value of each expression. Do not use a calculator. a. sec 18 tan 18 cos 20 b. cot 20 sin 20 c. tan 200 cot 20 25p p d . sin csc 12 12 2 2 Given One Value of a Trig Function, Find the Remaining Ones • Given that tan θ=½ and sin θ < 0, find the exact value of each of the remaining five trig functions of θ. • Using Definition • Using Fundamental Identities Even and Odd Properties sin( ) sin csc( ) csc cos( ) cos sec( ) sec tan( ) tan cot( ) cot Properties of Trig Functions • On-line Examples • On-line Tutorial