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2 電気電子数学入門サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行当時のものです.

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9 3 sin, cos, tan 5.1 x OP O OP r l θ θ = l/r [rad] ( ) l l = πr (πr)/r = π = π [rad] 1 = π/180 [rad] π/180

10 [rad] = 180 /π 180/π [rad] 5.1 (1) 150 = 150 π 180 = 5 6 π [rad] () 7 6 π [rad] = 7 6 π 180 π = 10 (0 <θ<π/) θ r 5. x y () ( ) () sin θ = y r, cos θ = x r, tan θ = y x (x 0) 5. sin θ, cosθ, tanθ θ ( )θ 5.3 θ r P (x, y) sin θ, cosθ, tanθ sin θ = y r, cos θ = x r, tan θ = y x (x 0) (5.1) θ tan θ x =0 θ

11 34 5 ( 1) r =1 ( ) 5.4 y sin θ x cos θ θ 0 θ π/ 5.1 θ 5.3 θ θ 0(0 ) sin θ 0 cos θ 1 tan θ 0 π 6 (30 ) 1 3 π 4 (45 ) π 3 (60 ) π (90 ) 1 0 θ sin θ + + cos θ + + tan θ + + π/ π/ + π/ π/ θ = π/ (5.1) θ sin θ, cosθ, tan θ tan θ = sin θ cos θ (5.) sin θ +cos θ =1 (5.3) (5.3) x + y = r n θ +nπ θ sin(θ +nπ) =sinθ cos(θ +nπ) =cosθ (5.4) tan(θ +nπ) =tanθ

12 (a) θ θ x sin( θ) = sin θ cos( θ) =cosθ (5.5) tan( θ) = tan θ (b) θ θ + π sin(θ + π) = sin θ cos(θ + π) = cos θ (5.6) tan(θ + π) =tanθ 5.5(c) θ θ + π/ ( sin θ + π ) =cosθ ( cos θ + π ) = sin θ (5.7) ( tan θ + π ) = 1 tan θ 5. (5.4) (5.7) (1) sin ( 53 ) ( ) 5 ( π = sin 3 π = sin π π ) ( = sin π ) ( π ) =sin =

13 36 5 ( 1) ( ) 10 () cos 3 π =cos (4π 3 ) π =cos ( 3 ) π = cos ( 3 ) π + π (3) tan ( 133 ) π ( π ) = cos 3 ( ) 13 = tan 3 π = 1 ( = tan 4π + π ) = tan π 3 3 = 3 α β α + β α β α β sin(α ± β) =sinαcos β ± cos α sin β [ 1) ] cos(α ± β) =cosα cos β sin α sin β tan(α ± β) = tan α ± tan β 1 tan α tan β [ ] [ ] (5.8) /index.html 5.1 sin 75, cos 15, tan( 15 ) sin 75 =sin( )=sin45 cos 30 + cos 45 sin 30 = = cos 15 =cos(45 30 )=cos45 cos 30 +sin45 sin 30 = = tan( 15 )=tan(30 45 )= tan 30 tan 45 = (1/ 3) 1 1+tan30 tan 45 1+(1/ 3) 1 = 1 3 = (1 3)( 3 1) 3+1 ( 3+1)( 3 1) = 4+ 3 = + 3 1) +/

14 α = β sin α =sin(α + α) =sinαcos α +cosαsin α =sinαcos α (5.9) cos α =cos(α + α) =cosαcos α sin α sin α =cos α sin α (5.10) (5.10) (5.3) cos α =1 sin α cos α =(1 sin α) sin α =1 sin α (5.11) (5.10) sin α =1 cos α cos α =cos α (1 cos α)=cos α 1 (5.1) (5.11), (5.1) sin α = 1 cos α, cos α = 1+cosα (5.13) 5.3 cos π 8 = 1+cos(π/4) = 1+(1/ ) = +1 = cos α =3/5 0 <α<π/ sin α, tanα, sinα, cosα sin α +cos α =1sin α =1 cos α =1 9/5 = 16/5 α 1 sin α = 4 5, sin α tan α = cos α = 4 3 sin α =sinαcos α = = 4 5 cos α =cos α sin α = = 7 5

15 38 5 ( 1) sin α cos β = 1 {sin(α + β)+sin(α β)} cos α sin β = 1 {sin(α + β) sin(α β)} cos α cos β = 1 {cos(α + β)+cos(α β)} (5.14) sin α sin β = 1 {cos(α + β) cos(α β)} 5.3 (5.8) (5.14) 1 ( ) = 1 {sin(α + β)+sin(α β)} = 1 {(sin α cos β +cosαsin β)+(sinαcos β cos α sin β)} =sinα cos β ( ) 1 ( /soft/73471/index.html ) (5.14) α = (A + B)/, β =(A B)/ sin A +sinb =sin A + B sin A sin B =cos A + B cos A +cosb =cos A + B cos A cos B = sin A + B cos A B sin A B cos A B sin A B (5.15)

16 93 11 y = f(x) x a b x Δx Δx = b a y Δy Δy = f(b) f(a) x a b f(x) Δy f(b) f(a) = Δx b a (1.1) y = f(x) A(a, f(a)) B(b, f(b)) [a, b] 1) y = f(x) 1.1 f(x) =x x 4 x Δx Δx =4 = y = f(x) y Δy Δy = f(4) f() = 4 =1 1) a x b [a, b]

17 94 1 Δy Δx = 16 =8 [, 4] f(x) =x 8 y = f(x) x a a + h Δy f(a + h) f(a) f(a + h) f(a) = = Δx a + h a h (1.) a h 0 11 lim h 0 f(a + h) f(a) lim h 0 h (1.3) (1.3) f(x) x = a ( ) f Δy (a) = lim h 0 Δx = lim f(a + h) f(a) h 0 h (1.4) f (a) b = a + h B (a + h, f(a + h)) (1.) A B AB a h 0 B A AB 1. B A A f(x) A f (a) A(a, f(a))

18 f(x) =3x x = a f 3(a + h) 3a 6ah +3h (a) = lim = lim = lim (6a +3h) h 0 h h 0 h h 0 =6a y = f(x) x dy dx = lim f(x + h) f(x) h 0 h (1.5) dy/dx df (x)/dx, f (x),y 1.3 (1) f(x) =x f (x + h) x h (x) = lim = lim h 0 h t 0 h =1 () f(x) =x f (x + h) x xh + h (x) = lim = lim = lim (x + h) h 0 h h 0 h h 0 =x (3) f(x) =x 3 f (x + h) 3 x 3 x 3 +3x h +3xh + h 3 x 3 (x) = lim = lim h 0 h h 0 h = lim h 0 (3x +3xh + h ) =3x 1.3 (1) (3) f(x) =x r f (x) =rx r 1 (1.6) (1.6) r

19 96 1 f(x) =x 4 x + x f(x) =(x + 1)(x 3 ) f(x) =(x 1)/(x +1) ( ) (1.6) x n y = kf(x) y = kf (x) (1.7) y = f(x) ± g(x) y = f (x) ± g (x) [ ] (1.8) y = f(x)g(x) y = f (x)g(x)+f(x)g (x) (1.9) y = f(x) g(x) y = f (x)g(x) f(x)g (x) {g(x)} (1.10) f(x) =1 y = 1 g(x) y = g (x) {g(x)} (1.11) y = {f(x)} n y = n{f(x)} n 1 f (x) (n ) (1.1) (1.7) {kf(x)} kf(x + h) kf(x) f(x + h) f(x) = lim = k lim = kf (x) h 0 h h 0 h (1.8) {f(x)+g(x)} {f(x + h)+g(x + h)} {f(x)+g(x)} = lim h 0 h f(x + h) f(x) g(x + h) g(x) = lim + lim h 0 h h 0 h = f (x)+g (x) {f(x) g(x)} = f (x) g (x) (1.9) (1.1) /73471/index.html 1.4 f(x) =x 3 +5x x +5 f(x) =(x 3 +5x x +5) =(x 3 ) +(5x ) (x) +(5)

20 = 3x +5 x 1=6x +10x (1) y =(x + 1)(x 1) () y = x +1 x + (1) y =(x +1) (x 1) + (x +1) (x 1) =(x 1) + (x +1) =x 1+x +=4x +1 () y = (x +) (x +1) x = x x +4 (x +)(x 1) = (x +) (x +) (x +) y = f(u) u u = g(x) y = f(g(x)) y = f(g(x)) dy dx = dy du du dx dy dx = f (u) g (x) (1.13) f(g(x)) u = g(x) (1.13) f (u) g (x) 1.5 y =(ax + b) u = ax + b y = u dy/du =u, du/dx = a dy dx = dy du =u a =a(ax + b) du dx 1. (K) =0 (K ) (sin x) =cosx

21 98 1 (cos x) = sin x (sin ax) = a cos ax (cos ax) = a sin ax (e x ) = e x (e ±ax ) = ±ae ±ax (a ) [ ] (log x ) = 1 x (log f(x) ) = f (x) f(x) (a x ) = a x log a (a >0, a 1) 1. (cos x) = sin x (5.15) (cos x) = lim h 0 cos(x + h) cos x h (cos x) = lim h 0 sin(x + h/) sin h/ h (11.10) (cos x) sin(h/) = lim sin(x + h/) = 1 sin x = sin x h 0 h/ sin x (5.15) y = f(x) y = f (x) x f(x) f (x), y, d y dx, d dx f(x) y = x 3 f (x) =3x, f (x) =6x

23 180 3 (scalar) (vector ) () O P.1 O P OP A A OP OP A A 1 u ( u =1) A A = Au = A u (.1). A = B

24 PQ RS A A A () 0 0 OO AA 0 0( ).1.3 A = B, C = D.3.4(a) x y A x () i y j A A = ia x + ja y =(A x,a y ) (.) A x,a y x y A (A x,a y ) A A x i + A y j ia x + ja y.4(b) x y z 3 A z k A A = ia x + ja y + ka z =(A x,a y,a z ) (.3)

25 18.4 A ( ) A = A = A x + A y + A z (.4) A = ia x + ja y + ka z B = ib x + jb y + kb z A = B A x = B x, A y = B y, A z = B z. i = i1+j0+k0 i =(1, 0, 0) j =(0, 1, 0), k =(0, 0, 1).1 (1) A = i+j3 () B = i+j4+k4 (1) A = A = A x + A y = +3 = 13 () B = B = B x + B y + B z = =6 A B C D C = A + B = i(a x + B x )+j(a y + B y )+k(a z + B z ) =(A x + B x,a y + B y,a z + B z )=(C x,c y,c z ) (.5) D = A B = i(a x B x )+j(a y B y )+k(a z B z )

26 =(A x B x,a y B y,a z B z )=(D x,d y,d z ) (.6) C D.5 (.5 ).5.3 A = i3 j k4 B = i4+j3 k C = A + B D = A B C = A + B = i{3+( 4)} + j{( ) + 3} + k{( 4) + ( )} = i + j k6 D = A B = i{3 ( 4)} + j{( ) 3} + k{( 4) ( )} = i7 j5 k (.5) A + B = B + A (A + B)+C = A +(B + C) A +( A) =0, A + 0 = A (.6) A B = A +( B) A A = 0 A (m ) A

27 184 m>0 ma A m m =1 A m<0 ma A m m = 1 A m =0 ma m(na) =(mn)a (m + n)a = ma + na m(a + B) =ma + mb 0 (m ) 0 m0 = 0. (1) 3A +5B A 4B () 3(A + B) (4A +B) (3) 1 (A +B) 1 (A B) 3 (1) 3A +5B A 4B =(3 )A +(5 4)B = A + B () 3(A + B) (4A +B) =6A +3B 8A 4B = A B (3) 1 (A +B) 1 3 (A B) = 1 A + B 3 A B = 1 6 A B A, B θ (0 θ π) (.6 ) (.7) AB A B C C = AB = A B.6 C = AB = A B = A B cos θ = AB cos θ (.7) (.7)

28 θ 0 A B = AB cos 0 = AB (.8) A = B A A = AA cos 0 = A θ π/ A B = AB cos π = 0 (.9) A B = B A (.10) A (B + C) =A B + A C (.11) (ka) B = k(a B) =A (kb) (.1) i, j, k i i =1, j j =1, k k = 1 (.13) i j =0, j k =0, k i =0, j i =0, k j =0, i k = 0 (.14) (.7) C = AB = A B = A x B x + A y B y + A z B z (.15) (.7) (.15) θ cos θ = A B AB = C AB = A xb x + A y B y + A z B z AB ( ) θ =Cos 1 Ax B x + A y B y + A z B z AB (.16) (.17)

29 186.3 C = AB = A B = A x B x +A y B y +A z B z C = AB = A B =(ia x + ja y + ka z ) (ib x + jb y + kb z ) (.11) (.1) A x B x (i i)+a x B y (i j)+a x B z (i k)+a y B x (j i) + A y B y (j j)+a y B z (j k)+a z B x (k i)+a z B y (k j) + A z B z (k k) (.13) (.14) i, j, k A x B x 1+A x B y 0+A x B z 0+A y B x 0+A y B y 1+A y B z 0 + A z B x 0+A z B y 0+A z B z 1 = A x B x + A y B y + A z B z C = AB = A B = A x B x + A y B y + A z B z.4 A = i3 j k4 B = i4+j3 k θ C = A B =3 ( 4) + ( ) 3+( 4) ( ) = 1 6+8= 10 A = 3 +( ) +( 4) = 9, B = ( 4) +3 +( ) = 9 cos θ = C AB = 10 9 ( θ =Cos 1 10 ) A, B C = A B

30 EXCEL 30, 47, 13, 168, 197, 06 y , , , , , , 78 4 () 4 ( ) ( ) (conjucate complex number) 10, (determinant) , 18 9 (imaginary part) 9, , , , 105 6, , , 113 6, , 17 (real part) 9, 58 1, 17 4,

31 , () () , , 15 94, 99, , , , , , 13, , , (complex number)

32 , 00 13, 68, 185, , , , ( ) ,

33 010 Printed in Japan ISBN

微分積分サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.

微分積分サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです. 微分積分サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)