4 4. 4.. 5 5 0 A P P P X X X X +45 45 0 45 60 70 X 60 X 0 P P
4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P 0 0 + 60 = 90, 0 + 60 = 750 0 + 60 ( ) = 0 90 750 0 90 0 X
4.. 5 n P X α P α + 60 n 4. 60 00 40 040 60 00 780 C 0 0 ( ) 60 a a a a 4. ( ) 80 () () 60
6 4 (0 80 ) (0 ) (50 ) 5 6 (90 ) (60 ) 6 (0 ) (5 ) 4 (90 ) 4 (45 ) (80 ) 0 (0 ) (80 ) 0 (0 ) 7 6 4 5 6 5 4 4.4 () 0 () 5 () 40 7 4 (4) 70 (5) 0
4.. 7 D r θ( ) l S l = rθ, S = r θ S = lr θ θ r l = rθ θ S rθ r S = r θ = r θ = lr 4. 0 6 S S = 0 6 = 5 4.5 () 4 7 () 6 6
8 4 4.. sin θ cos θ tan θ θ θ A x θ P(x, ) r r P (x, ) r x r x r r θ θ r x r x sin θ = r, cos θ = x r, tan θ = x r θ θ θ P θ tan θ 4. 4 r r = P (, ) x = = sin 4 = r = = cos 4 = x r = = tan 4 = x = = r Q P Q P r r 4 r x
4.. 9 4.6 θ sin θ cos θ tan θ () θ = 5 4 () θ = 6 () θ = sin θ cos θ tan θ θ sin θ cos θ tan θ + + + + x + x + x 4 4 4
40 4 4.7 θ () sin θ < 0 cos θ > 0 () cos θ < 0 tan θ > 0 B θ P(a, b) sin θ cos θ P(a, b) sin θ = b = b, cos θ = a = a θ cos θ x sin θ sin θ P cos θ P x = sin θ = cos θ α = sin θ α 5 θ α = cos θ α 5 θ θ θ = sin θ = cos θ
4.. 4 sin(θ + ) = sin θ, cos(θ + ) = cos θ sin θ cos θ = sin θ = cos θ = sin θ = cos θ θ P(a, b) x = T(, m) tan θ T(, m) P b tan θ = b a = m = m θ a tan θ T = tan θ x α α 5 θ tan θ θ = = tan θ θ θ =
4 4 = tan θ θ =, θ = tan(θ + ) = tan θ tan θ = sin θ = cos θ = tan θ C 4.4 = sin ( θ ı ) = sin θ θ = sin ( ) θ ı 4 0 6 θ 5 7 6 7 6 = sin θ 40 θ = cos θ = sin ( θ + ) = cos θ = sin θ θ
4.8 () = cos ( ) θ 4.. 4 () = sin ( θ + 6 ) () = tan ( θ 4 )
44 4 4.5 = sin θ = sin θ θ = sin θ 5 7 θ = sin θ 4.9 () = cos θ () = sin θ
4.6 = sin θ θ = α sin θ θ = α 4.. 45 sin θ = sin θ α θ α θ = sin θ = sin θ 5 7 θ 4.6 sin θ sin θ k sin kθ cos kθ k tan kθ k sin(kθ+)=sin kθ sin k(θ+ k )=sin kθ 4.0 () = cos θ
46 4 () = sin θ () = tan θ f(x) 0 p f(x + p) = f(x) f(x) p f(x + p) = f((x + p) + p) = f(x + p) = f(x) p p p
4.. 47 4.. sin(θ + ) = sin θ tan(θ + ) = tan θ A tan θ = sin θ cos θ + tan θ = cos θ sin θ + cos θ = 4. θ cos θ = 5 sin θ tan θ sin θ = cos θ = ( ) = 6 5 5 θ sin θ < 0 6 sin θ = 5 = 4 5 tan θ = sin θ ( cos θ = 4 ) ( ) 5 5 ( = 4 ) ( 5 ) = 4 5 5 5 θ 5 x 4. θ 4 sin θ = cos θ tan θ
48 4 4. θ 4 tan θ = sin θ cos θ cos θ = + tan θ = + ( ) = 5 θ 4 cos θ > 0 cos θ = 5 = 5 θ x sin θ = tan θ cos θ = ( ) 5 = 5 4. θ tan θ = sin θ cos θ B 4. tan θ + tan θ = sin θ cos θ = sin θ cos θ + cos θ sin θ = sin θ sin θ cos θ + = sin θ + cos θ sin θ cos θ cos θ sin θ cos θ = sin θ cos θ tan θ + tan θ = sin θ cos θ tan θ = cos θ sin θ sin θ+cos θ=
4.. 49 4. () (sin θ + cos θ) + (sin θ cos θ) = () tan θ sin θ = tan θ sin θ 4. sin θ + cos θ = () sin θ cos θ () sin θ + cos θ () sin θ cos θ (sin θ + cos θ) () a + b = (a + b)(a ab + b ) () sin θ + cos θ = sin θ + sin θ cos θ + cos θ = 4 sin θ+cos θ= + sin θ cos θ = 4 sin θ cos θ = 8 () sin θ + cos θ=(sin θ + cos θ)(sin θ sin θ cos θ + cos θ) =(sin θ + cos θ)( sin θ cos θ) = { ( )} 8 = 8 = 6
50 4 4.4 sin θ + cos θ = () sin θ cos θ () sin θ + cos θ 4.5 sin θ cos θ = () sin θ cos θ () sin θ cos θ
4.. 5 C sin(θ + n) = sin θ cos(θ + n) = cos θ n tan(θ+n)=tan θ tan(θ + n) = tan θ 4.7 () sin = sin 7 = sin () tan 7 = tan 4 = tan = + 4 7 = + = f(x) f( x) = f(x) f( x) = f(x) f(a) = f(x) a a x f(a) = f(x) f(a) a a x sin( θ) = sin θ =sin θ cos( θ) = cos θ =cos θ tan( θ) = tan θ =tan θ ( 4.8 () sin ) = sin = ( () cos ) = cos ( ) = cos + 4 = cos =
5 4 4.6 ( () sin ) () cos ( 6 ) 6 () tan ( 94 ) ( sin θ + ) = cos θ ( 4 cos θ + ) = sin θ ( tan θ + ) = tan θ sin(θ + ) = sin θ cos(θ + ) = cos θ tan(θ + ) = tan θ θ + P(a, b) θ x R( b, a) θ+ θ P(a, b) x Q( a, b) a = cos θ b = sin θ a = cos θ b = sin θ sin(θ + ) = b = sin θ sin ( θ + ) = a = cos θ cos(θ + ) = a = cos θ cos ( θ + ) = b = sin θ
4.. 5 4..4 A 4.4 0 θ < sin θ + = 0 sin θ = θ P Q θ 0 θ < P Q x θ = 7 6, 6 P 6 Q 4.4 θ 5 θ = 7 6 + n, θ = + n (n 6 ) 4.7 0 θ < () sin θ = () cos θ + = 0 () sin θ + = 0
54 4 4.5 tan θ = θ x = 0 θ < x = θ = θ x θ = + n (n ) 4.8 () tan θ = () tan θ =
4.. 55 4. 0 θ < 5 sin θ cos θ + 4 = 0 cos θ = sin θ sin θ sin θ 5 sin θ ( sin θ) + 4 = 0 sin θ + 5 sin θ + = 0 ( sin θ + )(sin θ + ) = 0 sin θ sin θ + = 0 sin θ+>0 0 θ < sin θ = θ = 7 6, 6 4.4 4.9 0 θ < () cos θ + 5 sin θ 4 = 0 () sin θ + cos θ = 0
56 4 B 4.6 0 θ < cos θ 0 θ < cos θ = θ θ = 4, 7 4 x 7 4 4 θ 7 4 4 4.6 = = cos θ = 4 = cos θ 0 θ < cos θ > 7 4 θ 0 θ < 4, 7 4 < θ < 4.0 0 θ < () sin θ > () cos θ
4.. 57 () tan θ < 4..5 () (sin θ + cos θ) + ( sin θ cos θ) = () cos(θ + ) + cos( θ) = () sin θ = () cos θ =
58 4 = sin θ + sin θ = sin θ + sin θ () sin θ = x = a(x p) + q () θ () 5 () 0 () θ = 4 + n θ = 5 + n (n ) () θ = 4 + n θ = 5 + n (n 4 ) () = (x + ) ()
4.. 59 4. 4.. α β sin(α + β) cos(α + β) α β tan(α + β) A P sin(α+β) sin(α + β) = HK = HQ + QK HQ = PQ cos β = sin α cos β QK = Q sin β = cos α sin β P H β Q α β S K x sin(α + β) = sin α cos β + cos α sin β P x cos(α + β) cos(α + β) = S = K PH K = Q cos β = cos α cos β, PH = PQ sin β = sin α sin β cos(α + β) = cos α cos β sin α sin β α β 4. β β sin(α β) = sin α cos β cos α sin β cos(α β) = cos α cos β + sin α sin β
60 4 sin(α + β) = sin α cos β + cos α sin β sin(α β) = sin α cos β cos α sin β cos(α + β) = cos α cos β sin α sin β 4 cos(α β) = cos α cos β + sin α sin β 75 5 4.9 () sin 75 =sin(45 + 0 ) =sin 45 cos 0 + cos 45 sin 0 = + = + = ( + ) = 6 + 4 () sin 5 =sin(45 0 ) =sin 45 cos 0 cos 45 sin 0 = 6 = = 4 4. () cos 75 () cos 5
4.. 6 B 5 tan(α + β) = 6 tan(α β) = tan(α + β) = tan(α + β) = sin(α + β) cos(α + β) tan α + tan β tan α tan β tan α tan β + tan α tan β sin α cos β + cos α sin β cos α cos β sin α sin β cos α cos β tan(α + β) = sin α cos α + sin β cos β sin α cos α sin β cos β = tan α + tan β tan α tan β 4.0 5 β β 6 tan 75 = tan(45 + 0 ) = tan 45 + tan 0 tan 45 tan 0 = + = + = ( + ) ( )( + ) = + + ( = 4 + ) = + 4. () tan 05 () tan 5
6 4 4.7 tan α = tan β = tan(α β) α β 0 < α β < 90 tan(α β) = tan α tan β + tan α tan β = + ( ) = 0 < α β < 90 tan(α β) = α β α β = 45 4.4 tan α = tan β = tan(α + β) α + β 0 < α + β < 80 x = mx θ tan θ = m tan θ = m = x = x x α β tan α = tan β = θ θ = α β α β 4.7 45 = x θ = mx θ α β x = x x
4.. 6 4.. sin(α + β) = sin α cos β + cos α sin β cos(α + β) = cos α cos β sin α sin β sin α cos α sin α sin α A β = α sin α = sin α cos α cos α = cos α sin α cos α = sin α cos α = cos α cos α= sin α sin α= cos α sin α 4.8 = tan α + cos α sin α cos α = + ( cos α ) sin α + cos α = sin α cos α cos α = sin α cos α = tan α = tan α 4.5 () (sin α + cos α) = + sin α
64 4 () cos 4 α sin 4 α = cos α cos α = sin α cos α = α cos α α sin α = cos α, cos α = + cos α 4. cos 8 ( cos 8 = ( + cos ) = ) + = + 4 4 cos 8 > 0 cos 8 = + + = 4 4.6 sin 8
4.. 65 tan α = tan α tan α, α tan = cos α + cos α 4.7 () tan α = tan α () 0 < α < cos α = tan α
66 4 4. 0 θ < cos θ + cos θ + = 0 cos θ ( cos θ ) + cos θ + = 0 cos θ( cos θ + ) = 0 cos θ = 0 cos θ = 0 θ < cos θ = 0 θ =, cos θ = θ = 4 ( ) θ = 4 4.8 0 θ < () cos θ + sin θ = () sin θ + cos θ = 0
4.. 67 B a sin θ + b cos θ sin θ + ( cos θ = sin θ + ) ( = sin θ cos + cos θ sin ) ( ) = sin θ + cos θ = sin θ + cos θ = a sin θ + b cos θ r sin(θ + α) (a, b) P x P α P = r b P(a, b) r r cos α = a, r sin α = b α a x a sin θ + b cos θ = r cos α sin θ + r sin α cos θ = r(sin θ cos α + cos θ sin α) = r sin(θ + α) r = a + b a sin θ + b cos θ a sin θ + b cos θ = a + b sin(θ + α) cos α = a a + b, sin α = b a + b
68 4 4. () sin θ + cos θ = ( sin θ + ) 4 ( () sin θ cos θ = sin θ ) 6 () () 4 P(, ) x 6 P(, ) x 4.9 r sin(θ + α) < α < () sin θ + cos θ () sin θ cos θ 4.9 = sin x + cos x sin x + cos x = ( sin x + ) 4 = ( sin x + ) 4 ( sin x + ) 4 4.()
4.. 69 = ( sin x + ) 4 = ( sin x + ) 4 4 4 4 = sin x 5 4 7 4 9 4 x = sin x x 4 4.0 = sin x + cos x
70 4 4.4 0 x < sin x cos x = r sin(x + α) x + α sin(x + α) x + α ( sin x ) = 6 ( sin x ) = 6 0 x < 6 x 6 < 6 x 6 = 6 x 6 = 5 6 x =, 4.4 0 x < = sin x cos x = x x
4.. 7 4. 0 x < sin x + cos x = 4.. 4 () sin α = sin α 4 sin α () cos α = cos α + 4 cos α
7 4 5 0 θ < () sin θ sin θ () cos θ < sin θ + 6 = sin x cos x
4.. 7 7 = sin x + cos x 4 α = α + α 5 () 0 θ, θ 5 () 0 < θ <, 7 6 < θ < 6 6 4 4 4 5 7 4 4 9 4 4 x 7 5 5
74 4 4. 4.. A () sin 6 () cos 7 ( () tan ) 6 () = cos θ ( () = sin θ + )
4.. 75 () tan θ sin θ sin θ tan θ = sin θ tan θ () cos θ tan θ = tan θ sin θ 4 0 x < () cos x sin x = 0 () cos x = cos x
76 4 5 ( ) () sin θ = cos θ () cos( θ) = cos θ 6 α β 4 sin α = 5 cos β = 4 5 () cos α () sin β () sin(α + β) (4) sin(α β) (5) cos(α + β) (6) cos(α β)
4.. 77 7 = sin x cos x (0 x < ) () () = 0 x () 0 x 4.. B 8 0 θ < = cos θ + sin θ θ
78 4 9 () sin α + sin β = cos α + cos β = cos(α β) () tan α = tan β = 4 tan γ = tan(α + β + γ) 0 sin(α β) sin(α + β) = tan α tan β tan α + tan β
4.. 79 () = sin x () = cos x () sin x + sin x = sin x cos x () 0 x < sin x + sin x = 0
80 4 = sin x cos x sin x + (0 x ) 9 () cos α cos β sin α sin β tan{(α + β) + γ} () () sin x = sin(x + x) a sin θ + b cos θ () () 0 ()
4.. 8 () θ () 5 7 6 6 6 5 6 8 9 6 θ [ () = cos θ cos θ () = cos θ sin θ sin θ cos θ 4 () x = 6, 5 6, () x = 0,, 4 ( ) 5 () sin θ =sin cos θ cos sin θ = cos θ 0 sin θ = cos θ () cos( θ)=cos cos θ + sin sin θ = cos θ + 0 sin θ = cos θ 6 () 4 5 () 5 () 4 5 (4) 0 (5) 7 5 7 () () x =, 4 () 0 x, 4 x < [ = sin (6) ( x ) ] ] 8 θ = sin θ = x x = (x ) + 9 () 59 7 () [ () tan(α + β) = 6 7 ] + β) + tan γ tan(α tan(α + β) tan γ
8 4 0 sin(α β) sin(α + β) = sin α cos β cos α sin β sin α cos β + cos α sin β cos α cos β sin(α β) sin(α + β) = sin α cos α sin β cos β sin α cos α + sin β cos β = tan α tan β tan α + tan β () 4 4 4 5 4 x [ () = + cos x ] x () sin x = sin(x + x) = sin x cos x + cos x sin x sin x = sin(x x) = sin x cos x cos x sin x () x = 0,,, () () sin x = 0 cos x = 0 x = 8 x = 5 [ 8 = (x sin + ) ] 4