SECTION 6.3 Properties of the Trigonometric Functions 417 Figure 43 Signs of the trigonometric functions y x (a) II (–, +) I (+, +) sin u . 0, csc u . 0, others negative All positive tan u . 0, cot u . 0, others negative cos u . 0, sec u . 0, others negative III (–, –) IV (+, –) y x (b) + + – – y x – + – + y x – + + – sine cosecant cosine secant tangent cotangent Finding the Quadrant in Which an Angle θ Lies If θ < sin 0 and θ < cos 0, name the quadrant in which the angle θ lies. Solution EXAMPLE 2 Let ( ) = P x y , be the point on the unit circle corresponding to θ. Then θ = < y sin 0 and θ = < x cos 0. Because points in quadrant III have < x 0 and < y 0 , θ lies in quadrant III. Now Work PROBLEM 2 7 4 Find the Values of the Trigonometric Functions Using Fundamental Identities If ( ) = P x y , is the point on the unit circle corresponding to θ, then θ θ θ θ θ θ = = = ≠ = ≠ = ≠ = ≠ y x y x x y y x x x y y sin cos tan if 0 csc 1 if 0 sec 1 if 0 cot if 0 Based on these definitions, we have the reciprocal identities : Two other fundamental identities are the quotient identities . The proofs of identities (2) and (3) follow from the definitions of the trigonometric functions. (See Problems 131 and 132.) If θ sin and θ cos are known, identities (2) and (3) make it easy to find the values of the remaining trigonometric functions. Reciprocal Identities θ θ θ θ θ θ = = = csc 1 sin sec 1 cos cot 1 tan (2) Quotient Identities θ θ θ θ θ θ = = tan sin cos cot cos sin (3)
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