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Section 6.3 Properties of the Trigonometric Functions If sin > 0 and cos < 0, name the quadrant in which the angle lies. For sin > 0 the y value must be positive so the angle must be in quadrant I or II. For cos < 0 the x value must be negative so the angle must be in quadrant II or III. Therefore, this angle must lie in quadrant II. 10 3 10 and cos , find the value of each of the four remaining 10 10 trigonometric functions of . Given sin 10 sin 10 10 1 10 tan cos 3 10 10 3 10 3 10 1 1 10 csc 10 sin 10 10 10 1 1 cot 3 tan 1 3 1 1 10 10 sec cos 3 10 3 10 3 10 Find the exact value of each expression. Do not use a calculator. 1 2 (a) cos 35 2 csc 35 cos 3 (b) cot 3 sin 3 1 2 2 2 (a) cos 35 sin 35 cos 35 1 2 csc 35 cos 3 (b) cot cot cot 0 3 3 3 sin 3 2 Given that sin and cos , find the exact value of each 5 of the remaining five trigonometric functions of . 2 Since sin 0 and cos 0, is in quadrant II. 5 The circle has a radius of 5 so its equation is x2 y 2 52 r 5 2 2 2 csc = P is a point on the circle so 5 x 2 y x 2 21 x 21 P(x,2) r=5 θ x 21 cos = r 5 2 r 5 5 21 sec = x 21 21 y 2 2 21 tan = x 21 21 x 21 cot = y 2 2 Given that sin and is an acute angle, find the exact value of each 5 of the remaining five trigonometric functions of . sin cos 1 2 2 2 2 2 co s 1 5 4 21 2 cos 1 25 25 cos 21 21 25 5 Since θ is in quadrant II, x values are negative 2 sin 2 5 2 21 5 tan cos 21 21 5 21 5 1 1 5 1 21 csc cot sin 2 2 tan 2 5 sec 1 cos 1 5 5 21 21 21 21 5 1 Given that cot and sin , find the exact value of each 3 of the remaining five trigonometric functions of . 1 Since cot 0 and sin 0, is in quadrant III. 3 2 2 2 The circle has a radius of 3 so its equation is x y 3 r 10 2 csc = P is a point on the circle so r 1 9 10 y r 10 θ P(-1,-3) 3 r 10 sec = 10 x 1 y 3 tan = 3 x 1 y 3 3 10 x 1 10 sin = cos = r 10 10 r 10 10 1 Given that cot and sin , find the exact value of each 3 of the remaining five trigonometric functions of . 1 1 3 3 10 sin cot 1 csc csc 10 10 10 2 3 1 1 1 2 tan 3 1 csc cot 1 3 3 1 10 3 10 2 csc 1 sin 3 10 1 10 10 9 9 cos tan 3 10 3 10 10 10 csc 1 1 10 sec 10 9 3 cos 10 10 Since sin θ < 0, csc θ is negative. 10 2 2 Find the exact value of: (a) cos (60°) (b) sin (390°) (c) 37 tan 4 1 (a) cos 60 cos 60 2 1 (b) sin 390 sin 390 sin 30 360 sin 30 2 37 (c) tan 4 37 tan 4 36 tan 4 4 tan 9 tan 1 4 4