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受欢迎的三角函数 >

3cos(x)+3sin(x)tan(x)=6

解答

3 cos (x) + 3 sin (x) tan (x) = 6

解答

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

+1

度数

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

3 cos (x) + 3 sin (x) tan (x) = 6

两边减去

6

3 cos (x) + 3 sin (x) tan (x) - 6 = 0

使用三角恒等式改写

- 6 + 3 cos (x) + 3 sin (x) tan (x)

使用基本三角恒等式:

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

= - 6 + 3 cos (x) + 3 sin (x) \frac{sin ( x )}{cos ( x )}

3 sin (x) \frac{sin ( x )}{cos ( x )} = \frac{3 sin ^{2} ( x )}{cos ( x )}

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

分式相乘:

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

= \frac{sin ( x ) \cdot 3 sin ( x )}{cos ( x )}

sin (x) \cdot 3 sin (x) = 3 sin^{2} (x)

sin (x) \cdot 3 sin (x)

使用指数法则:

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

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

= 3 sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= 3 sin^{2} (x)

= \frac{3 sin ^{2} ( x )}{cos ( x )}

= - 6 + 3 cos (x) + \frac{3 sin ^{2} ( x )}{cos ( x )}

使用毕达哥拉斯恒等式:

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

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

= - 6 + \frac{3 ( 1 - cos ^{2} ( x ) )}{cos ( x )} + 3 cos (x)

合并分式

\frac{3 ( - cos ^{2} ( x ) + 1 )}{cos ( x )} + 3 cos (x) : \frac{3}{cos ( x )}

\frac{3 ( - cos ^{2} ( x ) + 1 )}{cos ( x )} + 3 cos (x)

将项转换为分式:

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

= \frac{3 ( 1 - cos ^{2} ( x ) )}{cos ( x )} + \frac{3 cos ( 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{3 ( 1 - cos ^{2} ( x ) ) + 3 cos ( x ) cos ( x )}{cos ( x )}

3 (1 - cos^{2} (x)) + 3 cos (x) cos (x) = 3 (1 - cos^{2} (x)) + 3 cos^{2} (x)

3 (1 - cos^{2} (x)) + 3 cos (x) cos (x)

3 cos (x) cos (x) = 3 cos^{2} (x)

3 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)

= 3 cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= 3 cos^{2} (x)

= 3 (- cos^{2} (x) + 1) + 3 cos^{2} (x)

= \frac{3 ( - cos ^{2} ( x ) + 1 ) + 3 cos ^{2} ( x )}{cos ( x )}

乘开

3 (1 - cos^{2} (x)) + 3 cos^{2} (x) : 3

3 (1 - cos^{2} (x)) + 3 cos^{2} (x)

乘开

3 (1 - cos^{2} (x)) : 3 - 3 cos^{2} (x)

3 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 3, b = 1, c = cos^{2} (x)

= 3 \cdot 1 - 3 cos^{2} (x)

数字相乘:

3 \cdot 1 = 3

= 3 - 3 cos^{2} (x)

= 3 - 3 cos^{2} (x) + 3 cos^{2} (x)

同类项相加:

- 3 cos^{2} (x) + 3 cos^{2} (x) = 0

= 3

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

= \frac{3}{cos ( x )} - 6

- 6 + \frac{3}{cos ( x )} = 0

- 6 + \frac{3}{cos ( x )} = 0

在两边乘以

cos (x)

- 6 + \frac{3}{cos ( x )} = 0

在两边乘以

cos (x)

- 6 cos (x) + \frac{3}{cos ( x )} cos (x) = 0 \cdot cos (x)

化简

- 6 cos (x) + \frac{3}{cos ( x )} cos (x) = 0 \cdot cos (x)

化简

\frac{3}{cos ( x )} cos (x) : 3

\frac{3}{cos ( x )} cos (x)

分式相乘:

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

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

约分:

cos (x)

= 3

化简

0 \cdot cos (x) : 0

0 \cdot cos (x)

使用法则

0 \cdot a = 0

= 0

- 6 cos (x) + 3 = 0

- 6 cos (x) + 3 = 0

- 6 cos (x) + 3 = 0

将

3

到右边

- 6 cos (x) + 3 = 0

两边减去

3

- 6 cos (x) + 3 - 3 = 0 - 3

化简

- 6 cos (x) = - 3

- 6 cos (x) = - 3

两边除以

- 6

- 6 cos (x) = - 3

两边除以

- 6

\frac{- 6 cos ( x )}{- 6} = \frac{- 3}{- 6}

化简

\frac{- 6 cos ( x )}{- 6} = \frac{- 3}{- 6}

化简

\frac{- 6 cos ( x )}{- 6} : cos (x)

\frac{- 6 cos ( x )}{- 6}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{6 cos ( x )}{6}

数字相除:

\frac{6}{6} = 1

= cos (x)

化简

\frac{- 3}{- 6} : \frac{1}{2}

\frac{- 3}{- 6}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{3}{6}

约分:

3

= \frac{1}{2}

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

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

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

验证解

找到无定义的点（奇点）:

cos (x) = 0

取

- 6 + \frac{3}{cos ( x )}

的分母，令其等于零

cos (x) = 0

以下点无定义

cos (x) = 0

将不在定义域的点与解相综合:

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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

作图

流行的例子

tan^{2} (2 x) - 1 = 0

3sin(4x)=-6sin(2x)

3 sin (4 x) = - 6 sin (2 x)

2sin(2θ)-sqrt(3)=0

2 sin (2 θ) - 3 = 0

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

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

4cos(2θ)+12=-2cos(θ)+9

4 cos (2 θ) + 12 = - 2 cos (θ) + 9