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受欢迎的三角函数 >

tan(2x)sin(x)=sin(x)

解答

tan (2 x) sin (x) = sin (x)

解答

x = 2 πn, x = π + 2 πn, x = \frac{π}{8} + \frac{πn}{2}

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 22. 5^{\circ} + 9 0^{\circ} n

求解步骤

tan (2 x) sin (x) = sin (x)

两边减去

sin (x)

tan (2 x) sin (x) - sin (x) = 0

分解

tan (2 x) sin (x) - sin (x) : sin (x) (tan (2 x) - 1)

tan (2 x) sin (x) - sin (x)

因式分解出通项

sin (x)

= sin (x) (tan (2 x) - 1)

sin (x) (tan (2 x) - 1) = 0

分别求解每个部分

sin (x) = 0 or tan (2 x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0

的通解

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

tan (2 x) - 1 = 0 : x = \frac{π}{8} + \frac{πn}{2}

tan (2 x) - 1 = 0

将

1

到右边

tan (2 x) - 1 = 0

两边加上

1

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

化简

tan (2 x) = 1

tan (2 x) = 1

tan (2 x) = 1

的通解

tan (x)

周期表（周期为

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

2 x = \frac{π}{4} + πn

2 x = \frac{π}{4} + πn

解

2 x = \frac{π}{4} + πn : x = \frac{π}{8} + \frac{πn}{2}

2 x = \frac{π}{4} + πn

两边除以

2

2 x = \frac{π}{4} + πn

两边除以

2

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

化简

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

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{π}{4}}{2} + \frac{πn}{2} : \frac{π}{8} + \frac{πn}{2}

\frac{\frac{π}{4}}{2} + \frac{πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

使用分式法则:

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

= \frac{π}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{π}{8}

= \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

合并所有解

x = 2 πn, x = π + 2 πn, x = \frac{π}{8} + \frac{πn}{2}

作图

流行的例子

sin(u)=-3/5 ,(3pi)/2 <u<2pi

sin (u) = - \frac{3}{5}, \frac{3 π}{2} < u < 2 π

2-2sin^2(x/2)=sin^2(x)

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

3cos(B)-1=5cos(B)+1

3 cos (B) - 1 = 5 cos (B) + 1

cos(θ)=(sqrt(3))/4

cos (θ) = \frac{3}{4}

cos (x) = \frac{- 3}{5}