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受欢迎的三角函数 >

(cos(x))/(sin(2x))= 5/7

解答

\frac{cos ( x )}{sin ( 2 x )} = \frac{5}{7}

解答

x = 0.77539 \dots + 2 πn, x = π - 0.77539 \dots + 2 πn

+1

度数

x = 44.42700 \dots^{\circ} + 36 0^{\circ} n, x = 135.57299 \dots^{\circ} + 36 0^{\circ} n

求解步骤

\frac{cos ( x )}{sin ( 2 x )} = \frac{5}{7}

两边减去

\frac{5}{7}

\frac{cos ( x )}{sin ( 2 x )} - \frac{5}{7} = 0

化简

\frac{cos ( x )}{sin ( 2 x )} - \frac{5}{7} : \frac{7 cos ( x ) - 5 sin ( 2 x )}{7 sin ( 2 x )}

\frac{cos ( x )}{sin ( 2 x )} - \frac{5}{7}

sin (2 x), 7

的最小公倍数:

7 sin (2 x)

sin (2 x), 7

最小公倍数 (LCM)

计算出由出现在

sin (2 x)

或

7

中的因子组成的表达式

= 7 sin (2 x)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

7 sin (2 x)

对于

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

将分母和分子乘以

7

\frac{cos ( x )}{sin ( 2 x )} = \frac{cos ( x ) \cdot 7}{sin ( 2 x ) \cdot 7}

对于

\frac{5}{7} :

将分母和分子乘以

sin (2 x)

\frac{5}{7} = \frac{5 sin ( 2 x )}{7 sin ( 2 x )}

= \frac{cos ( x ) \cdot 7}{sin ( 2 x ) \cdot 7} - \frac{5 sin ( 2 x )}{7 sin ( 2 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{cos ( x ) \cdot 7 - 5 sin ( 2 x )}{7 sin ( 2 x )}

\frac{7 cos ( x ) - 5 sin ( 2 x )}{7 sin ( 2 x )} = 0

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

7 cos (x) - 5 sin (2 x) = 0

使用三角恒等式改写

- 5 sin (2 x) + 7 cos (x)

使用倍角公式:

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

= - 5 \cdot 2 sin (x) cos (x) + 7 cos (x)

化简

= - 10 sin (x) cos (x) + 7 cos (x)

7 cos (x) - 10 cos (x) sin (x) = 0

分解

7 cos (x) - 10 cos (x) sin (x) : - cos (x) (10 sin (x) - 7)

7 cos (x) - 10 cos (x) sin (x)

因式分解出通项

- cos (x)

= - cos (x) (- 7 + 10 sin (x))

- cos (x) (10 sin (x) - 7) = 0

分别求解每个部分

cos (x) = 0 or 10 sin (x) - 7 = 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

10 sin (x) - 7 = 0 : x = arcsin (\frac{7}{10}) + 2 πn, x = π - arcsin (\frac{7}{10}) + 2 πn

10 sin (x) - 7 = 0

将

7

到右边

10 sin (x) - 7 = 0

两边加上

7

10 sin (x) - 7 + 7 = 0 + 7

化简

10 sin (x) = 7

10 sin (x) = 7

两边除以

10

10 sin (x) = 7

两边除以

10

\frac{10 sin ( x )}{10} = \frac{7}{10}

化简

sin (x) = \frac{7}{10}

sin (x) = \frac{7}{10}

使用反三角函数性质

sin (x) = \frac{7}{10}

sin (x) = \frac{7}{10}

的通解

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

x = arcsin (\frac{7}{10}) + 2 πn, x = π - arcsin (\frac{7}{10}) + 2 πn

x = arcsin (\frac{7}{10}) + 2 πn, x = π - arcsin (\frac{7}{10}) + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (\frac{7}{10}) + 2 πn, x = π - arcsin (\frac{7}{10}) + 2 πn

因为方程对以下值无定义:

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

x = arcsin (\frac{7}{10}) + 2 πn, x = π - arcsin (\frac{7}{10}) + 2 πn

以小数形式表示解

x = 0.77539 \dots + 2 πn, x = π - 0.77539 \dots + 2 πn

作图

流行的例子

tan(x)=(6*0.809)/(15)

tan (x) = \frac{6 \cdot 0.809}{15}

arctan (x) = 26

tan (θ) = - \frac{8}{6}

cos^2(θ)-3/2 = 5/2 cos(θ)

cos^{2} (θ) - \frac{3}{2} = \frac{5}{2} cos (θ)

tan (x) = - \frac{1}{8}