1 (1) ( i ) 60 (ii) 75 (iii) 15 () ( i ) (ii) 4 (iii) 7 1 ( () r, AOB = θ 0 < θ < ) OAB A OB P ( AB ) < ( AP ) (4) 0 < θ < sin θ < θ < tan θ 0 x, 0 y (1) sin x = sin y (x, y) () cos x cos y (x, y) 1 c
(1) 0 θ 1080 sin 6θ = sin 15θ = 0 θ () (1) θ sin θ sin(θ 0 ) 4 (1) : cos( θ) = cos θ, sin( θ) = sin θ () : cos( θ) = cos θ, sin( θ) = sin θ ( ) ( ) () : cos θ = sin θ, sin θ = cos θ c
5 (1) cos(θ + 55 ) (0 θ 180 ) () sin θ < cos θ (0 θ < ) 6 (1) θ a sin θ + b cos θ + c = 0 a, b, c () 0 x <, 0 y < { cos x + sin y = 1 sin x + cos y = c
7 sin θ + cos θ = 1 (1) sin θ cos θ () sin θ + cos θ () tan θ + 1 tan θ 8 (1) cos(α + β) = cos α cos β sin α sin β sin(α + β) = sin α cos β + cos α sin β tan α + tan β tan(α + β) = 1 tan α tan β () α, β sin α = 1, sin β = sin(α + β), tan(α + β) 6 5 1 4 c
9 (1) θ = 7 cos θ = cos θ, sin θ = sin θ () cos 7 10 (1) ABC cos A + cos B + cos C = 1 + 4 sin A sin B sin C () ABC A = 60 sin B sin C 5 c
11 (1) < θ < < sin θ cos θ < () 0 θ sin θ + 4 cos θ 1 (1) y = 4 sin x + cos x + cos x () y = ( sin x + 1)( cos x + 1) () y = sin x + sin x cos x + cos x ( 0 x ) 6 c
1 x + y = 0, x y + = 0 14 1 C 1, (4, 0) C C 1 A(1, 0) P C B(6, 0) Q P Q PQ = AB P, Q 7 c
15 A(4, ) O 60 B 16 (1) ( i ) y = cos x (ii) y = sin x (iii) y = sin ( x ) 4 () cos θ 1 sin θ () z z 8 c
1 1 60 60 r r 1 ( ) (rad) ( ) θ, l, S l = rθ, S = r θ = 1 r θ 180 = (rad) () () 0 < θ < = sin θ < θ < tan θ ( III ) (1) (i) (ii) 5 1 (iii) 7 4 () (i) 45 (ii) 10 (iii) 105 () 0 < θ < OAB OAP 1 r θ < 1 r r tan θ B rθ < r tan θ θ AB rθ, AP r tan θ O ( AB ) < ( AP ) () (4) ( OAB ) < ( OAB ) < ( OAP ) 1 r sin θ < 1 r θ < 1 r r tan θ sin θ < θ < tan θ () P A 9 c
x + y = 1 (1, 0) y 0 θ (cos θ, sin θ) θ (cos θ, sin θ), (0, 0), (cos θ, sin θ) O 1 x tan θ x, y (1) : sin α = sin β β = α + n β = α + n cos α = cos β β = ± α + n (n ) (1) x y y = x + n y = x + n (n ) 0 x n, 0 y n Y O x Y =sin x 1 X y O x () x cos x cos y y Y 0 x Y x x O 1 X x O 1 X X=cos x X=cos x 0 x 0 y x x y x 0 y x x y ( ) y O x 10 c
(cos θ, sin θ) θ (60 ) cos θ, sin θ tan θ ( 180 ) f(x) 0 p f(x + p) = f(x) f(x) p p 1 (1) sin 6θ = 0, sin 15θ = 0 6θ = 180 m, 15θ = 180 n θ = 0 m = 1 n 5m = n (m, n ) 5 m = k, n = 5k (k ) (m, n ) θ = (0m) = (60k), 0 θ 1080 θ = (60k) (k = 0, 1,,,, 18) () f(θ) = sin θ sin(θ 0 ) sin f(θ + 180 ) = sin θ{ sin(θ 0 )} = sin θ sin(θ 0 ) = f(θ) f(θ) 180 f(0 ) = 0 f(60 ) = sin 60 sin 0 = f(10 ) = sin 10 sin 90 = 18 k=0 f ( (60k) ) = 0 7 + 1 = 1 = 4 4 6 + 6 = 9 (cos(θ + 180 ), sin(θ + 180 )), (cos θ, sin θ) (cos(θ + 180 ), sin(θ + 180 )) = ( cos θ, sin θ) f(θ) 180 f(180 ), f(40 ), f(00 ) ( 10 ) sin θ sin(θ 0 ) = 1 { cos 0 cos(θ 0 ) } 11 c
4 θ θ θ ( ) ( ) θ + ( θ) (1) θ = (cos θ, sin θ) (cos( θ), sin( θ)) y (cos( θ), sin( θ)) = ( cos θ, sin θ) () () (cos θ, sin θ), (cos( θ), sin( θ)) x (cos( θ), sin( θ)) = (cos θ, sin θ) () ( ) θ + θ () θ = 4 ( ( ) ( )) (cos θ, sin θ) cos θ, sin θ y = x ( ( ) ) cos θ, sin( ) θ = (sin θ, cos θ) () ( 8 ) 1 c
5 cos θ a sin θ b (1) x + y = 1 x () x + y = 1 y < x x + y x + y (1) 0 θ 180 55 θ + 55 415 x + y = 1, x 15 θ + 55 405 10 θ 175 y O 1 x () x + y = 1 y < x y O 1 x 0 θ < 4 5 < θ < 4 1 c
6 cos θ sin θ cos θ + sin θ = 1 (1) θ a, b () cos θ sin θ () x, y cos x + sin x = 1, cos y + sin y = 1 4 4 cos x, sin x, cos y, sin y (1) θ a sin θ + b cos θ + c = 0 θ = 0 b + c = 0 θ = a + c = 0 θ = b + c = 0 1 a = b = c = 0 0 sin θ + 0 cos θ + 0 = 0 θ a = b = c = 0 () cos x + sin y = 1 1 sin x + cos y = cos x + sin x = 1 cos y + sin y = 1 4 1, 4 y ( sin x) + (1 cos x) = 1 sin x + cos x sin x cos x + 4 = 1 sin x + cos x = 5, 5 cos x ( sin x) + sin x = 1 4 sin x 4 sin x + = ( sin x ) = 0 5 sin x =, cos x = 1 1, cos y =, sin y = 1 0 x <, 0 y < x =, y = 6 14 c
7 x, y f(x, y) f(x, y) = f(y, x) f(x, y) ( ) f(x, y) x + y, xy cos θ sin θ cos θ + sin θ = 1 cos θ + sin θ, cos θ sin θ cos θ + sin θ, cos θ sin θ (1) sin θ + cos θ = 1 1 sin θ + sin θ cos θ + cos θ = 1 4 sin θ + cos θ = 1 sin θ cos θ = 8 () 1, sin θ + cos θ = (sin θ + cos θ) sin θ cos θ (sin θ + cos θ) ( 1 ) ( = ) 1 8 = 11 16 () tan θ = sin θ cos θ () tan θ + 1 tan θ = sin θ cos θ + cos θ sin θ = sin θ + cos θ sin θ cos θ = 8 () sin θ + cos θ = (sin θ + cos θ)(sin θ sin θ cos θ + cos θ) = 1 { ( 1 )} 8 = 11 16 15 c
8 (i) (ii) III (cos α + i sin α)(cos β + i sin β) = cos(α + β) + i sin(α + β) tan sin, cos () (1) (1, 0), (0, 1) θ (1, 0) (cos θ, sin θ) ( (0, 1) cos ( + θ), sin ( + θ)) = ( sin θ, cos θ) 4 (0, 0), (cos α, 0), (cos α, sin α), (0, sin α) β (cos α, sin α) = (cos α) (1, 0) + (sin α) (0, 1) (cos α) (cos β, sin β) + (sin α) ( sin β, cos β) (cos α, sin α) (cos(α + β), sin(α + β)) cos(α + β) = cos α cos β sin α sin β sin(α + β) = sin α cos β + cos α sin β cos(α + β) 0, cos α 0, cos β 0 sin(α + β) sin α cos β + cos α sin β tan(α + β) = = cos(α + β) cos α cos β sin α sin β sin α cos α + sin β cos β tan α + tan β () sin α = ( 0 < α < ), sin β = 1 = 1 sin α cos α 6 5 1 sin β cos β = ( 0 < β < 1 tan α tan β ) cos α =, cos β = 17 1 5 1, tan α =, tan β = 6 17 sin(α + β) = 17 1 5 1 + 6 1 5 1 = 4 5 + 6 17 tan(α + β) = 1 = 4 6 17 () 16 c
9 8 α = β cos α = cos α sin α, sin α = sin α cos α cos α + sin α = 1 cos α = cos α 1 = 1 sin α cos α = cos(α + α), sin α = sin(α + α) cos α = 4 cos α cos α, sin α = sin α 4 sin α (i) (ii) sin α cos α θ = 5 (1) θ = 7 5θ = 60 θ = 60 θ 60 cos θ = cos θ, sin θ = sin θ () () sin θ = sin θ sin θ 4 sin θ = sin θ cos θ sin θ( 4 sin θ + cos θ) = 0 sin θ = sin 7 0 4 sin θ + cos θ = 0 sin θ + cos θ = 1 4(1 cos θ) + cos θ = 0 4 cos θ + cos θ 1 = 0 cos θ = cos 7 > 0 cos 7 = 1 + 5 4 cos θ = cos θ cos θ 17 c
10 ( 8 ) sin α cos β = 1 {sin(α + β) + sin(α β)} cos α cos β = 1 {cos(α + β) + cos(α β)} sin α sin β = 1 {cos(α β) cos(α + β)} α + β = A, α β = B sin A + sin B = sin A+B cos A B cos A + cos B = cos A+B cos A cos B = sin A+B cos A B sin A B A = A + B + A B, B = A + B A B () (1) cos A + cos B = cos A + B cos A B A + B + C = 180 cos C 1 = cos(a + B) 1 = cos A + B cos A + cos B + cos C 1 = cos A + B ( cos A B ( = sin 90 C () B + C = 10 ( ) sin B sin C = 1 {cos(b C) cos(b + C)} = 1 0 < B < 10 10 < B 10 < 10 sin B sin C 0 < sin B sin C 4 cos A + B ) sin A sin B = 4 sin A sin B sin C () ) { cos(b 10 ) + 1 } 18 c
11 sin x cos α + cos x sin α = sin(x + α) cos x cos α + sin x sin α = cos(x α) ( ) 1 a sin x + b cos x (a, b 0 ) a + b = 1 a = cos α, b = sin α ( a = sin α, b = cos α ) a sin x + b cos x = ( a + b a a + b sin x + b ) a + b cos x ( a ) + ( b ) = 1 a + b a + b sin cos ( (1) sin θ cos θ = sin θ 1 ) cos θ = ( sin θ cos 6 cos θ sin 6 = sin ( θ ) 6 < sin ( θ ) < 6 ) y 60 O 1 y = y = x < θ < < θ 6 < 5 6 < θ 6 < 4 5 6 < θ < 11 1 ( () sin θ + 4 cos θ = 5 5 sin θ + 4 ) 5 cos θ = 5(sin θ cos α + cos θ sin α) α cos α = 5, sin α = 4 5, 0 < α < α < θ + α < + α = 5 sin(θ + α) θ + α = θ + α = + α 5, 4 α cos α, sin α 19 c
1 ( i ) (ii) sin x, cos x, tan x, sin x + cos x t t (iii) a sin x + b cos x 1 (1) 4 sin x + cos x + cos x = 4(1 cos x) + cos x + ( cos x 1) ( = cos x + cos x + = cos x 1 1 cos x 1 cos x = 1 7, 1 () sin x + cos x = t t = ( 1 sin x + 1 ) cos x ) + 7 cos x = 1 = ( sin x cos 4 + cos x sin ) ( ) = sin x + 4 4 t t t = sin x + cos x sin x + cos x = 1 t = 1 + sin x cos x sin x cos x = t 1 y = 4 sin x cos x + (sin x + cos x) + 1 = (t 1) + t + 1 ( = t + 1 ) t t = t = 1 +, () y = sin x + sin x cos x + cos 1 cos x 1 + cos x x = + sin x + = sin x + cos x + () sin x + cos x = = sin ( x + ) 4 4 x + 4 5 4 x + 4 = x + 4 = 5 4 +, 1 0 c
1 tan tan tan α, tan β tan 90 = tan α tan β = 1 90 < α < 90, 90 < β < 90 α β, β α, 180 (α β), 180 (β α) ( 0 90 ) tan 1 4 tan tan(β α) = tan(α β) tan{180 (α β)} = tan(α β) tan{180 (β α)} = tan(α β) () tan α, tan β θ tan α tan β = 1 θ = 90 tan α tan β 1 tan θ = tan(α β) tan α = 1, tan β = 1 ( 90 < α < 90, 90 < β < 90 ) tan α tan β = 1 6 1 θ ( 0 < θ < 90 ) tan θ = tan(α β) tan α tan β = 1 + tan α tan β 1 = 1 1 1 1 = 1 0 < θ < 90 θ = 45 45 1 c
14 cos θ, sin θ Q θ P θ sin θ + cos θ = 1 P(cos θ, sin θ), Q(4 + cos θ, sin θ) PQ = (cos θ cos θ 4) + (sin θ sin θ) = (cos θ cos θ) 8(cos θ cos θ) + 16 + (sin θ sin θ) = (cos θ cos θ) + (sin θ sin θ) 8 cos θ + 16 cos θ + 16 = 5 4(cos θ cos θ + sin θ sin θ) 8 cos θ + 16 cos θ + 16 = 5 4 cos(θ θ) 8( cos θ 1) + 16 cos θ + 16 = 16 cos θ + 1 cos θ + 9 PQ = AB = 5 16 cos θ + 1 cos θ + 9 = 5 16 cos θ 1 cos θ 4 = 0 4(cos θ 1)(4 cos θ + 1) = 0 0 < θ < θ cos θ = 1 15 4, sin θ = 4 ( cos θ = 1 ) 7 15 1 =, sin θ = 4 8 4 P, Q ( P 7 15 ) ( 7 15 ) 8,, Q 8, ( 1 ) 15 = 4 8 c
15 III ( 8 ) r (r cos θ, r sin θ) φ (r cos(θ + φ), r sin(θ + φ)) (4, ) = (r cos θ, r sin θ) (r > 0) r = 4 + = 5, cos θ = 4 5, sin θ = 5 B (5 cos(θ + 60 ), 5 sin(θ + 60 )) cos(θ + 60 ) = cos θ cos 60 sin θ sin 60 = 4 1 5 = 4 5 10 sin(θ + 60 ) = sin θ cos 60 + cos θ sin 60 = 1 5 + 4 = + 4 5 10 ( 4 B, + 4 ) c
16 y = sin x y 1 O 1 5, 11, 1 ( ) II III II (1) () (1) ( i ) cos x = sin ( x + ) ( ) 8 y = cos x y = sin x x y 1 x O 1 x (ii) y = sin x y = sin x x 1 y 1 1 O 4 4 5 4 7 4 x 4 c
(iii) y = sin ( x ) y = sin x x 4 4 y 1 1 1 O 4 4 5 4 7 4 11 4 x () cos θ 1 sin θ 0 θ y = cos θ y = 1 sin θ z 1 1 O 1 α cos θ = 1 sin θ β { y = max cos θ, θ ( ) 1 } sin θ θ α, β { 1 } z = max cos θ, sin θ cos β ( ) sin θ + cos θ = 1 (= 1 sin β ) ( cos θ) + cos θ = 1 cos θ = 1 5 θ < θ < β cos β = 1 5 = z 5 5 5 5 5 c