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受欢迎的三角函数 >

5tan(x)sin(x)=6sin(x)

解答

5 tan (x) sin (x) = 6 sin (x)

解答

x = 2 πn, x = π + 2 πn, x = 0.87605 \dots + πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 50.19442 \dots^{\circ} + 18 0^{\circ} n

求解步骤

5 tan (x) sin (x) = 6 sin (x)

两边减去

6 sin (x)

5 tan (x) sin (x) - 6 sin (x) = 0

分解

5 tan (x) sin (x) - 6 sin (x) : sin (x) (5 tan (x) - 6)

5 tan (x) sin (x) - 6 sin (x)

因式分解出通项

sin (x)

= sin (x) (5 tan (x) - 6)

sin (x) (5 tan (x) - 6) = 0

分别求解每个部分

sin (x) = 0 or 5 tan (x) - 6 = 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

5 tan (x) - 6 = 0 : x = arctan (\frac{6}{5}) + πn

5 tan (x) - 6 = 0

将

6

到右边

5 tan (x) - 6 = 0

两边加上

6

5 tan (x) - 6 + 6 = 0 + 6

化简

5 tan (x) = 6

5 tan (x) = 6

两边除以

5

5 tan (x) = 6

两边除以

5

\frac{5 tan ( x )}{5} = \frac{6}{5}

化简

tan (x) = \frac{6}{5}

tan (x) = \frac{6}{5}

使用反三角函数性质

tan (x) = \frac{6}{5}

tan (x) = \frac{6}{5}

的通解

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{6}{5}) + πn

x = arctan (\frac{6}{5}) + πn

合并所有解

x = 2 πn, x = π + 2 πn, x = arctan (\frac{6}{5}) + πn

以小数形式表示解

x = 2 πn, x = π + 2 πn, x = 0.87605 \dots + πn

作图

流行的例子

2tan^3(x)-6tan(x)=0

2 tan^{3} (x) - 6 tan (x) = 0

cos(4x)(2cos(x)+1)=0

cos (4 x) (2 cos (x) + 1) = 0

cos (3 x) - 1 = 0

tan (α) = \frac{3}{4}

cos^2(x)-cos(x)-1=0,0<= x<= 2pi

cos^{2} (x) - cos (x) - 1 = 0, 0 \leq x \leq 2 π