sin
2
x
+
cos
2
x
=
1
\sin ^2 x + \cos ^2 x = 1
sin2x+cos2x=1
tan
x
=
sin
x
cos
x
\tan x = \dfrac{\sin x}{\cos x}
tanx=cosxsinx
cot
x
=
1
tan
x
=
cos
x
sin
x
\cot x = \dfrac{1}{\tan x}=\dfrac{\cos x}{\sin x}
cotx=tanx1=sinxcosx
sec
x
=
1
cos
x
\sec x= \dfrac{1}{\cos x}
secx=cosx1
csc
x
=
1
sin
x
\csc x =\dfrac{1}{\sin x}
cscx=sinx1
tan
2
x
=
sec
2
−
1
=
1
cos
2
x
−
1
=
1
−
cos
2
x
cos
2
x
=
sin
2
x
cos
2
x
\tan^2x=\sec^2-1=\dfrac{1}{\cos^2x}-1=\dfrac{1-\cos^2x}{\cos^2x}=\dfrac{\sin^2x}{\cos^2x}
tan2x=sec2−1=cos2x1−1=cos2x1−cos2x=cos2xsin2x
cot
2
=
csc
2
x
−
1
=
1
sin
2
x
−
1
=
1
−
sin
2
x
sin
2
x
=
cos
2
x
sin
2
x
\cot^2=\csc^2x-1=\dfrac{1}{\sin^2x}-1=\dfrac{1-\sin^2x}{\sin^2x}=\dfrac{\cos^2x}{\sin^2x}
cot2=csc2x−1=sin2x1−1=sin2x1−sin2x=sin2xcos2x
cos
x
=
sin
(
x
+
π
2
)
\cos x=\sin(x+\dfrac{\pi}{2})
cosx=sin(x+2π)，
sin
x
\sin x
sinx 向左平移
π
2
\dfrac{\pi}{2}
2π. (左加右减)
sin
x
=
cos
(
x
−
π
2
)
\sin x=\cos(x-\dfrac{\pi}{2})
sinx=cos(x−2π)
cos
x
=
cos
(
−
x
)
\cos x= \cos(-x)
cosx=cos(−x)，偶函数
sin
x
=
−
sin
(
−
x
)
\sin x = - \sin(-x)
sinx=−sin(−x)，奇函数
sin
x
=
−
sin
(
x
±
π
)
\sin x= -\sin(x\pm\pi)
sinx=−sin(x±π)，
sin
x
\sin x
sinx无论是向左、还是向右平移
π
\pi
π 个单位后，乘以-1，关于x轴对称之后函数图像不变.
cos
x
=
−
cos
(
x
±
π
)
\cos x = -\cos(x\pm\pi)
cosx=−cos(x±π)
arcsin
x
+
arccos
x
=
π
2
\arcsin x+\arccos x=\dfrac{\pi}{2}
arcsinx+arccosx=2π.
倍(半)角公式
cos
(
A
±
B
)
=
cos
A
⋅
cos
B
∓
sin
A
⋅
sin
B
\cos(A\pm B)=\cos A\cdot\cos B \mp \sin A\cdot\sin B
cos(A±B)=cosA⋅cosB∓sinA⋅sinB.
sin
(
A
±
B
)
=
sin
A
⋅
cos
B
±
cos
A
⋅
sin
B
\sin(A\pm B)=\sin A\cdot\cos B \pm \cos A\cdot\sin B
sin(A±B)=sinA⋅cosB±cosA⋅sinB.
cos
(
2
A
)
=
cos
2
A
−
sin
2
A
=
1
−
2
sin
2
A
=
2
cos
2
A
−
1
\cos(2A)=\cos^2A-\sin^2A=1-2\sin^2A=2\cos^2A-1
cos(2A)=cos2A−sin2A=1−2sin2A=2cos2A−1.
cos
A
=
cos
2
A
2
−
sin
2
A
2
=
1
−
2
sin
2
A
2
=
2
cos
2
A
2
−
1
\cos A = \cos^2\dfrac{A}{2}-\sin^2\dfrac{A}{2}=1-2\sin^2\dfrac{A}{2}=2\cos^2\dfrac{A}{2}-1
cosA=cos22A−sin22A=1−2sin22A=2cos22A−1.
sin
(
2
A
)
=
2
sin
A
⋅
cos
A
\sin(2A)=2\sin A\cdot\cos A
sin(2A)=2sinA⋅cosA.
sin
A
=
2
sin
A
2
⋅
cos
A
2
\sin A = 2\sin\dfrac{A}{2}\cdot\cos\dfrac{A}{2}
sinA=2sin2A⋅cos2A.
tan
2
α
=
2
tan
α
1
−
tan
2
α
\tan2\alpha=\dfrac{2\tan\alpha}{1-\tan^2\alpha}
tan2α=1−tan2α2tanα.
tan
α
=
2
tan
α
2
1
−
tan
2
α
2
\tan\alpha=\dfrac{2\tan\dfrac{\alpha}{2}}{1-\tan^2\dfrac{\alpha}{2}}
tanα=1−tan22α2tan2α.
tan
α
2
=
1
−
cos
α
sin
α
=
sin
α
1
+
cos
α
\tan\dfrac{\alpha}{2}=\dfrac{1-\cos\alpha}{\sin\alpha}=\dfrac{\sin\alpha}{1+\cos\alpha}
tan2α=sinα1−cosα=1+cosαsinα.
tan
2
α
=
sin
2
α
cos
2
α
=
2
sin
α
⋅
cos
α
1
−
2
sin
2
α
=
2
tan
α
1
cos
2
α
−
2
tan
2
α
=
2
tan
α
1
−
sin
2
α
cos
2
α
−
tan
2
α
=
2
tan
α
1
−
tan
2
α
.
\tan2\alpha=\dfrac{\sin2\alpha}{\cos2\alpha}=\dfrac{2\sin\alpha\cdot\cos\alpha}{1-2\sin^2\alpha}=\dfrac{2\tan\alpha}{\dfrac{1}{\cos^2\alpha}-2\tan^2\alpha}=\dfrac{2\tan\alpha}{\dfrac{1-\sin^2\alpha}{\cos^2\alpha}-\tan^2\alpha}=\dfrac{2\tan\alpha}{1-\tan^2\alpha}.
tan2α=cos2αsin2α=1−2sin2α2sinα⋅cosα=cos2α1−2tan2α2tanα=cos2α1−sin2α−tan2α2tanα=1−tan2α2tanα.
tan
α
=
2
tan
α
2
1
−
tan
2
α
2
\tan\alpha=\dfrac{2\tan\dfrac{\alpha}{2}}{1-\tan^2\dfrac{\alpha}{2}}
tanα=1−tan22α2tan2α
tan
α
−
tan
α
tan
2
α
2
=
2
tan
α
2
\tan\alpha-\tan\alpha\tan^2\dfrac{\alpha}{2}=2\tan\dfrac{\alpha}{2}
tanα−tanαtan22α=2tan2α
tan
α
⋅
tan
2
α
2
+
2
tan
α
2
−
tan
α
=
0
\tan\alpha\cdot\tan^2\dfrac{\alpha}{2}+2\tan\dfrac{\alpha}{2}-\tan\alpha=0
tanα⋅tan22α+2tan2α−tanα=0 求根公式：
tan
α
2
=
−
2
±
4
+
4
tan
2
α
2
tan
α
=
−
1
±
sec
α
tan
α
=
−
cos
α
±
1
sin
α
\tan\dfrac{\alpha}{2}=\dfrac{-2\pm\sqrt{4+4\tan^2\alpha}}{2\tan\alpha}=\dfrac{-1\pm\sec\alpha}{\tan\alpha}=\dfrac{-\cos\alpha\pm1}{\sin\alpha}
tan2α=2tanα−2±4+4tan2α
=tanα−1±secα=sinα−cosα±1 当
α
∈
(
0
,
π
)
\alpha\in(0,\pi)
α∈(0,π) 时，
tan
α
2
>
0
\tan\dfrac{\alpha}{2}>0
tan2α>0，而
−
cos
α
+
1
sin
α
<
0
-\dfrac{\cos\alpha+1}{\sin\alpha}<0
−sinαcosα+1<0.
∴
tan
α
2
=
−
cos
α
+
1
sin
α
\therefore \tan\dfrac{\alpha}{2}=-\dfrac{\cos\alpha+1}{\sin\alpha}
∴tan2α=−sinαcosα+1 不成立.
∴
tan
α
2
=
1
−
cos
α
sin
α
=
(
1
−
cos
α
)
⋅
(
1
+
cos
α
)
sin
α
+
sin
α
⋅
cos
α
=
1
−
cos
2
α
sin
α
+
sin
α
⋅
cos
α
=
sin
α
1
+
cos
α
\therefore \tan\dfrac{\alpha}{2}=\dfrac{1-\cos\alpha}{\sin\alpha}=\dfrac{(1-\cos\alpha)\cdot(1+\cos\alpha)}{\sin\alpha+\sin\alpha\cdot\cos\alpha}=\dfrac{1-\cos^2\alpha}{\sin\alpha+\sin\alpha\cdot\cos\alpha}=\dfrac{\sin\alpha}{1+\cos\alpha}
∴tan2α=sinα1−cosα=sinα+sinα⋅cosα(1−cosα)⋅(1+cosα)=sinα+sinα⋅cosα1−cos2α=1+cosαsinα
正、余弦化切弦
sin
2
x
=
2
sin
x
⋅
cos
x
=
2
tan
x
sec
2
x
=
2
tan
x
1
+
tan
2
x
\sin2x=2\sin x\cdot\cos x=\dfrac{2\tan x}{\sec^2x}=\dfrac{2\tan x}{1+\tan^2x}
sin2x=2sinx⋅cosx=sec2x2tanx=1+tan2x2tanx.
cos
2
x
=
cos
2
x
−
sin
2
x
=
1
−
tan
2
x
sec
2
x
=
1
−
tan
2
x
1
+
tan
2
x
\cos2x=\cos^2x-\sin^2x=\dfrac{1-\tan^2x}{\sec^2x}=\dfrac{1-\tan^2x}{1+\tan^2x}
cos2x=cos2x−sin2x=sec2x1−tan2x=1+tan2x1−tan2x.
sin
x
=
2
tan
x
2
1
+
tan
2
x
2
\sin x=\dfrac{2\tan\dfrac{x}{2}}{1+\tan^2\dfrac{x}{2}}
sinx=1+tan22x2tan2x.
cos
x
=
1
−
tan
2
x
2
1
+
tan
2
x
2
\cos x=\dfrac{1-\tan^2\dfrac{x}{2}}{1+\tan^2\dfrac{x}{2}}
cosx=1+tan22x1−tan22x.
辅助角公式
sin
α
⋅
a
a
2
+
b
2
−
cos
α
⋅
b
a
2
+
b
2
=
sin
α
⋅
cos
β
−
cos
α
⋅
sin
β
=
sin
(
α
−
β
)
\sin\alpha\cdot\dfrac{a}{\sqrt{a^2+b^2}}-\cos\alpha\cdot\dfrac{b}{\sqrt{a^2+b^2}}=\sin\alpha\cdot\cos\beta-\cos\alpha\cdot\sin\beta=\sin(\alpha-\beta)
sinα⋅a2+b2
a−cosα⋅a2+b2
b=sinα⋅cosβ−cosα⋅sinβ=sin(α−β).
其中，令
cos
β
=
a
a
2
+
b
2
\cos\beta=\dfrac{a}{\sqrt{a^2+b^2}}
cosβ=a2+b2
a，
sin
β
=
b
a
2
+
b
2
\sin\beta=\dfrac{b}{\sqrt{a^2+b^2}}
sinβ=a2+b2
b，
则有
cos
2
β
+
sin
2
β
=
(
a
a
2
+
b
2
)
2
+
(
b
a
2
+
b
2
)
2
=
1
\cos^2\beta+\sin^2\beta=\Big(\dfrac{a}{\sqrt{a^2+b^2}}\Big)^2+\Big(\dfrac{b}{\sqrt{a^2+b^2}}\Big)^2=1
cos2β+sin2β=(a2+b2
a)2+(a2+b2
b)2=1.
E
m
L
2
ω
2
+
R
2
⋅
(
R
⋅
sin
ω
t
−
L
ω
⋅
cos
ω
t
)
\dfrac{E_m}{L^2\omega^2+R^2}\cdot\big(R\cdot\sin\omega t-L\omega\cdot\cos\omega t\big)
L2ω2+R2Em⋅(R⋅sinωt−Lω⋅cosωt)
=
E
m
L
2
ω
2
+
R
2
⋅
(
sin
ω
t
⋅
R
L
2
ω
2
+
R
2
−
cos
ω
t
⋅
L
ω
L
2
ω
2
+
R
2
)
=\dfrac{E_m}{\sqrt{L^2\omega^2+R^2}}\cdot\big(\sin\omega t\cdot\dfrac{R}{\sqrt{L^2\omega^2+R^2}}-\cos\omega t\cdot\dfrac{L\omega}{\sqrt{L^2\omega^2+R^2}}\big)
=L2ω2+R2
Em⋅(sinωt⋅L2ω2+R2
R−cosωt⋅L2ω2+R2
Lω)
=
E
m
L
2
ω
2
+
R
2
⋅
sin
(
ω
t
−
φ
)
=\dfrac{E_m}{\sqrt{L^2\omega^2+R^2}}\cdot\sin(\omega t-\varphi)
=L2ω2+R2
Em⋅sin(ωt−φ).
其中
cos
φ
=
R
L
2
ω
2
+
R
2
\cos\varphi=\dfrac{R}{\sqrt{L^2\omega^2+R^2}}
cosφ=L2ω2+R2
R，
sin
φ
=
L
ω
L
2
ω
2
+
R
2
\sin\varphi=\dfrac{L\omega}{\sqrt{L^2\omega^2+R^2}}
sinφ=L2ω2+R2
Lω.
积化和差、和差化积
参考往期文章，点击跳转