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受欢迎的三角函数 >

5tan(x)=9sin(x)

解答

5 tan (x) = 9 sin (x)

解答

x = 2 πn, x = π + 2 πn, x = 0.98176 \dots + 2 πn, x = 2 π - 0.98176 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 56.25101 \dots^{\circ} + 36 0^{\circ} n, x = 303.74898 \dots^{\circ} + 36 0^{\circ} n

求解步骤

5 tan (x) = 9 sin (x)

两边减去

9 sin (x)

5 tan (x) - 9 sin (x) = 0

用 sin, cos 表示

5 tan (x) - 9 sin (x)

使用基本三角恒等式:

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

= 5 \cdot \frac{sin ( x )}{cos ( x )} - 9 sin (x)

化简

5 \cdot \frac{sin ( x )}{cos ( x )} - 9 sin (x) : \frac{5 sin ( x ) - 9 sin ( x ) cos ( x )}{cos ( x )}

5 \cdot \frac{sin ( x )}{cos ( x )} - 9 sin (x)

乘

5 \cdot \frac{sin ( x )}{cos ( x )} : \frac{5 sin ( x )}{cos ( x )}

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

分式相乘:

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

= \frac{sin ( x ) \cdot 5}{cos ( x )}

= \frac{5 sin ( x )}{cos ( x )} - 9 sin (x)

将项转换为分式:

9 sin (x) = \frac{9 sin ( x ) cos ( x )}{cos ( x )}

= \frac{sin ( x ) \cdot 5}{cos ( x )} - \frac{9 sin ( x ) cos ( 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{sin ( x ) \cdot 5 - 9 sin ( x ) cos ( x )}{cos ( x )}

= \frac{5 sin ( x ) - 9 sin ( x ) cos ( x )}{cos ( x )}

\frac{5 sin ( x ) - 9 cos ( x ) sin ( x )}{cos ( x )} = 0

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

5 sin (x) - 9 cos (x) sin (x) = 0

分解

5 sin (x) - 9 cos (x) sin (x) : - sin (x) (9 cos (x) - 5)

5 sin (x) - 9 cos (x) sin (x)

因式分解出通项

- sin (x)

= - sin (x) (- 5 + 9 cos (x))

- sin (x) (9 cos (x) - 5) = 0

分别求解每个部分

sin (x) = 0 or 9 cos (x) - 5 = 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

9 cos (x) - 5 = 0 : x = arccos (\frac{5}{9}) + 2 πn, x = 2 π - arccos (\frac{5}{9}) + 2 πn

9 cos (x) - 5 = 0

将

5

到右边

9 cos (x) - 5 = 0

两边加上

5

9 cos (x) - 5 + 5 = 0 + 5

化简

9 cos (x) = 5

9 cos (x) = 5

两边除以

9

9 cos (x) = 5

两边除以

9

\frac{9 cos ( x )}{9} = \frac{5}{9}

化简

cos (x) = \frac{5}{9}

cos (x) = \frac{5}{9}

使用反三角函数性质

cos (x) = \frac{5}{9}

cos (x) = \frac{5}{9}

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{5}{9}) + 2 πn, x = 2 π - arccos (\frac{5}{9}) + 2 πn

x = arccos (\frac{5}{9}) + 2 πn, x = 2 π - arccos (\frac{5}{9}) + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn, x = arccos (\frac{5}{9}) + 2 πn, x = 2 π - arccos (\frac{5}{9}) + 2 πn

以小数形式表示解

x = 2 πn, x = π + 2 πn, x = 0.98176 \dots + 2 πn, x = 2 π - 0.98176 \dots + 2 πn

作图

流行的例子

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8tan(θ/2)+8cos(θ)tan(θ/2)=1

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6 cos^{2} (x) + cos (x) - 1 = 0

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