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受欢迎的三角函数 >

5sin^2(x)=9tan(x)

解答

5 sin^{2} (x) = 9 tan (x)

解答

x = 2 πn, x = π + 2 πn

+1

度数

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

求解步骤

5 sin^{2} (x) = 9 tan (x)

两边减去

9 tan (x)

5 sin^{2} (x) - 9 tan (x) = 0

用 sin, cos 表示

5 sin^{2} (x) - 9 tan (x)

使用基本三角恒等式:

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

= 5 sin^{2} (x) - 9 \cdot \frac{sin ( x )}{cos ( x )}

化简

5 sin^{2} (x) - 9 \cdot \frac{sin ( x )}{cos ( x )} : \frac{5 sin ^{2} ( x ) cos ( x ) - 9 sin ( x )}{cos ( x )}

5 sin^{2} (x) - 9 \cdot \frac{sin ( x )}{cos ( x )}

乘

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

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

分式相乘:

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

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

= 5 sin^{2} (x) - \frac{9 sin ( x )}{cos ( x )}

将项转换为分式:

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

= \frac{5 sin ^{2} ( x ) cos ( x )}{cos ( x )} - \frac{sin ( x ) \cdot 9}{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{5 sin ^{2} ( x ) cos ( x ) - sin ( x ) \cdot 9}{cos ( x )}

= \frac{5 sin ^{2} ( x ) cos ( x ) - 9 sin ( x )}{cos ( x )}

\frac{- 9 sin ( x ) + 5 cos ( x ) sin ^{2} ( x )}{cos ( x )} = 0

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

- 9 sin (x) + 5 cos (x) sin^{2} (x) = 0

分解

- 9 sin (x) + 5 cos (x) sin^{2} (x) : sin (x) (- 9 + 5 sin (x) cos (x))

- 9 sin (x) + 5 cos (x) sin^{2} (x)

使用指数法则:

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

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

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

因式分解出通项

sin (x)

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

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

分别求解每个部分

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

无解

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

使用三角恒等式改写

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

使用倍角公式:

2 sin (x) cos (x) = sin (2 x)

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

= - 9 + 5 \cdot \frac{sin ( 2 x )}{2}

- 9 + 5 \cdot \frac{sin ( 2 x )}{2} = 0

将

9

到右边

- 9 + 5 \cdot \frac{sin ( 2 x )}{2} = 0

两边加上

9

- 9 + 5 \cdot \frac{sin ( 2 x )}{2} + 9 = 0 + 9

化简

5 \cdot \frac{sin ( 2 x )}{2} = 9

5 \cdot \frac{sin ( 2 x )}{2} = 9

整理

5 \cdot \frac{sin ( 2 x )}{2} : \frac{5 sin ( 2 x )}{2}

5 \cdot \frac{sin ( 2 x )}{2}

分式相乘:

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

= \frac{sin ( 2 x ) \cdot 5}{2}

\frac{5 sin ( 2 x )}{2} = 9

在两边乘以

2

\frac{5 sin ( 2 x )}{2} = 9

在两边乘以

2

\frac{5 sin ( 2 x )}{2} \cdot 2 = 9 \cdot 2

化简

\frac{5 sin ( 2 x )}{2} \cdot 2 = 9 \cdot 2

化简

\frac{5 sin ( 2 x )}{2} \cdot 2 : 5 sin (2 x)

\frac{5 sin ( 2 x )}{2} \cdot 2

分式相乘:

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

= \frac{5 sin ( 2 x ) \cdot 2}{2}

约分:

2

= 5 sin (2 x)

化简

9 \cdot 2 : 18

9 \cdot 2

数字相乘:

9 \cdot 2 = 18

= 18

5 sin (2 x) = 18

5 sin (2 x) = 18

5 sin (2 x) = 18

两边除以

5

5 sin (2 x) = 18

两边除以

5

\frac{5 sin ( 2 x )}{5} = \frac{18}{5}

化简

sin (2 x) = \frac{18}{5}

sin (2 x) = \frac{18}{5}

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = 2 πn, x = π + 2 πn

作图

流行的例子

cos (θ) = \frac{5}{4}

csc^2(x)=4cot^2(x)

csc^{2} (x) = 4 cot^{2} (x)

3 cos (θ) = 2, 1

3 csc (x) - 7 = 0

tan(2θ)=0.6583

tan (2 θ) = 0.6583