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受欢迎的三角函数 >

sin(x)tan(x)=-2-cot(x)sin(x)

解答

sin (x) tan (x) = - 2 - cot (x) sin (x)

解答

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

+1

度数

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

求解步骤

sin (x) tan (x) = - 2 - cot (x) sin (x)

两边减去

- 2 - cot (x) sin (x)

sin (x) tan (x) + 2 + cot (x) sin (x) = 0

用 sin, cos 表示

2 + cot (x) sin (x) + sin (x) tan (x)

使用基本三角恒等式:

cot (x) = \frac{cos ( x )}{sin ( x )}

= 2 + \frac{cos ( x )}{sin ( x )} sin (x) + sin (x) tan (x)

使用基本三角恒等式:

tan (x) = \frac{sin ( x )}{cos ( x )}

= 2 + \frac{cos ( x )}{sin ( x )} sin (x) + sin (x) \frac{sin ( x )}{cos ( x )}

化简

2 + \frac{cos ( x )}{sin ( x )} sin (x) + sin (x) \frac{sin ( x )}{cos ( x )} : \frac{2 cos ( x ) + cos ^{2} ( x ) + sin ^{2} ( x )}{cos ( x )}

2 + \frac{cos ( x )}{sin ( x )} sin (x) + sin (x) \frac{sin ( x )}{cos ( x )}

\frac{cos ( x )}{sin ( x )} sin (x) = cos (x)

\frac{cos ( x )}{sin ( x )} sin (x)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{cos ( x ) sin ( x )}{sin ( x )}

约分:

sin (x)

= cos (x)

sin (x) \frac{sin ( x )}{cos ( x )} = \frac{sin ^{2} ( x )}{cos ( x )}

sin (x) \frac{sin ( x )}{cos ( x )}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( x ) sin ( x )}{cos ( x )}

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sin^{2} (x)

= \frac{sin ^{2} ( x )}{cos ( x )}

= 2 + cos (x) + \frac{sin ^{2} ( x )}{cos ( x )}

将项转换为分式:

2 = \frac{2 cos ( x )}{cos ( x )}, cos (x) = \frac{cos ( x ) cos ( x )}{cos ( x )}

= \frac{2 cos ( x )}{cos ( x )} + \frac{cos ( x ) cos ( x )}{cos ( x )} + \frac{sin ^{2} ( x )}{cos ( x )}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 cos ( x ) + cos ( x ) cos ( x ) + sin ^{2} ( x )}{cos ( x )}

2 cos (x) + cos (x) cos (x) + sin^{2} (x) = 2 cos (x) + cos^{2} (x) + sin^{2} (x)

2 cos (x) + cos (x) cos (x) + sin^{2} (x)

cos (x) cos (x) = cos^{2} (x)

cos (x) cos (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

= 2 cos (x) + cos^{2} (x) + sin^{2} (x)

= \frac{2 cos ( x ) + cos ^{2} ( x ) + sin ^{2} ( x )}{cos ( x )}

= \frac{2 cos ( x ) + cos ^{2} ( x ) + sin ^{2} ( x )}{cos ( x )}

\frac{cos ^{2} ( x ) + sin ^{2} ( x ) + 2 cos ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos^{2} (x) + sin^{2} (x) + 2 cos (x) = 0

使用三角恒等式改写

cos^{2} (x) + sin^{2} (x) + 2 cos (x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

= 2 cos (x) + 1

2 cos (x) + 1 = 0

将

1

到右边

2 cos (x) + 1 = 0

两边减去

1

2 cos (x) + 1 - 1 = 0 - 1

化简

2 cos (x) = - 1

2 cos (x) = - 1

两边除以

2

2 cos (x) = - 1

两边除以

2

\frac{2 cos ( x )}{2} = \frac{- 1}{2}

化简

cos (x) = - \frac{1}{2}

cos (x) = - \frac{1}{2}

cos (x) = - \frac{1}{2}

的通解

cos (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

作图

流行的例子

-2sin^2(x)+cos(x)(2+2cos(x))=0

- 2 sin^{2} (x) + cos (x) (2 + 2 cos (x)) = 0

tan (\frac{θ}{2}) = 2

sin (x) = 6

sin(2x)=5cos^2(x)

sin (2 x) = 5 cos^{2} (x)

sin^2(θ)+cos(θ)=2

sin^{2} (θ) + cos (θ) = 2