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受欢迎的三角函数 >

3cos(x)=sqrt(2)cot(x)

解答

3 cos (x) = 2 cot (x)

解答

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.49088 \dots + 2 πn, x = π - 0.49088 \dots + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 28.12550 \dots^{\circ} + 36 0^{\circ} n, x = 151.87449 \dots^{\circ} + 36 0^{\circ} n

求解步骤

3 cos (x) = 2 cot (x)

两边减去

2 cot (x)

3 cos (x) - 2 cot (x) = 0

用 sin, cos 表示

3 cos (x) - cot (x) 2

使用基本三角恒等式:

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

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

化简

3 cos (x) - \frac{cos ( x )}{sin ( x )} 2 : \frac{3 cos ( x ) sin ( x ) - 2 cos ( x )}{sin ( x )}

3 cos (x) - \frac{cos ( x )}{sin ( x )} 2

乘

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

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

分式相乘:

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

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

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

将项转换为分式:

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

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

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

\frac{- cos ( x ) 2 + 3 cos ( x ) sin ( x )}{sin ( x )} = 0

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

- cos (x) 2 + 3 cos (x) sin (x) = 0

分解

- cos (x) 2 + 3 cos (x) sin (x) : cos (x) (- 2 + 3 sin (x))

- cos (x) 2 + 3 cos (x) sin (x)

因式分解出通项

cos (x)

= cos (x) (- 2 + 3 sin (x))

cos (x) (- 2 + 3 sin (x)) = 0

分别求解每个部分

cos (x) = 0 or - 2 + 3 sin (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

cos (x) = 0

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}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

- 2 + 3 sin (x) = 0 : x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

- 2 + 3 sin (x) = 0

将

2

到右边

- 2 + 3 sin (x) = 0

两边加上

2

- 2 + 3 sin (x) + 2 = 0 + 2

化简

3 sin (x) = 2

3 sin (x) = 2

两边除以

3

3 sin (x) = 2

两边除以

3

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

化简

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

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

使用反三角函数性质

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

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

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

以小数形式表示解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.49088 \dots + 2 πn, x = π - 0.49088 \dots + 2 πn

作图

流行的例子

5 cos (7 θ) = 0

- 8 tan (y) = 0

2cos^2(θ)+cos(θ)=0,0<= θ<2pi

2 cos^{2} (θ) + cos (θ) = 0, 0 \leq θ < 2 π

sqrt(3)cos(2x)+sin(2x)=0

3 cos (2 x) + sin (2 x) = 0

cos((-pi)/2+x/2)=0

cos (\frac{- π}{2} + \frac{x}{2}) = 0