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受欢迎的三角函数 >

50/33 =(sin(x))/(sin(120-x))

解答

\frac{50}{33} = \frac{sin ( x )}{sin ( 12 0 ^{\circ} - x )}

解答

x = 1.38810 \dots + 18 0^{\circ} n

+1

弧度

x = 1.38810 \dots + πn

求解步骤

\frac{50}{33} = \frac{sin ( x )}{sin ( 12 0 ^{\circ} - x )}

交换两边

\frac{sin ( x )}{sin ( 12 0 ^{\circ} - x )} = \frac{50}{33}

使用三角恒等式改写

\frac{sin ( x )}{sin ( 12 0 ^{\circ} - x )} = \frac{50}{33}

使用三角恒等式改写

sin (12 0^{\circ} - x)

使用角差恒等式:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= sin (12 0^{\circ}) cos (x) - cos (12 0^{\circ}) sin (x)

化简

sin (12 0^{\circ}) cos (x) - cos (12 0^{\circ}) sin (x) : \frac{3}{2} cos (x) + \frac{1}{2} sin (x)

sin (12 0^{\circ}) cos (x) - cos (12 0^{\circ}) sin (x)

化简

sin (12 0^{\circ}) : \frac{3}{2}

sin (12 0^{\circ})

使用以下普通恒等式:

sin (12 0^{\circ}) = \frac{3}{2}

sin (x)

周期表（周期为

36 0^{\circ} n

"）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

= \frac{3}{2} cos (x) - cos (12 0^{\circ}) sin (x)

化简

cos (12 0^{\circ}) : - \frac{1}{2}

cos (12 0^{\circ})

使用以下普通恒等式:

cos (12 0^{\circ}) = - \frac{1}{2}

cos (x)

周期表（周期为

36 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - \frac{1}{2}

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

使用法则

- (- a) = a

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

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

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} = \frac{50}{33}

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} = \frac{50}{33}

两边减去

\frac{50}{33}

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} - \frac{50}{33} = 0

化简

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} - \frac{50}{33} : \frac{16 sin ( x ) - 50 3 cos ( x )}{33 ( 3 cos ( x ) + sin ( x ) )}

\frac{sin ( x )}{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )} - \frac{50}{33}

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

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

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

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

分式相乘:

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

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

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

\frac{1}{2} sin (x)

分式相乘:

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

= \frac{1 \cdot sin ( x )}{2}

乘以:

1 \cdot sin (x) = sin (x)

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

= \frac{sin ( x )}{\frac{3 c o s ( x )}{2} + \frac{s i n ( x )}{2}}

合并分式

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

使用法则

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

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

= \frac{sin ( x )}{\frac{3 c o s ( x ) + s i n ( x )}{2}}

使用分式法则:

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

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

= \frac{2 sin ( x )}{3 cos ( x ) + sin ( x )} - \frac{50}{33}

3 cos (x) + sin (x), 33

的最小公倍数:

33 (3 cos (x) + sin (x))

3 cos (x) + sin (x), 33

最小公倍数 (LCM)

计算出由出现在

3 cos (x) + sin (x)

或

33

中的因子组成的表达式

= 33 (3 cos (x) + sin (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

33 (3 cos (x) + sin (x))

对于

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

将分母和分子乘以

33

\frac{sin ( x ) \cdot 2}{3 cos ( x ) + sin ( x )} = \frac{sin ( x ) \cdot 2 \cdot 33}{( 3 cos ( x ) + sin ( x ) ) \cdot 33} = \frac{66 sin ( x )}{33 ( 3 cos ( x ) + sin ( x ) )}

对于

\frac{50}{33} :

将分母和分子乘以

3 cos (x) + sin (x)

\frac{50}{33} = \frac{50 ( 3 cos ( x ) + sin ( x ) )}{33 ( 3 cos ( x ) + sin ( x ) )}

= \frac{66 sin ( x )}{33 ( 3 cos ( x ) + sin ( x ) )} - \frac{50 ( 3 cos ( x ) + sin ( x ) )}{33 ( 3 cos ( x ) + 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{66 sin ( x ) - 50 ( 3 cos ( x ) + sin ( x ) )}{33 ( 3 cos ( x ) + sin ( x ) )}

乘开

66 sin (x) - 50 (3 cos (x) + sin (x)) : 16 sin (x) - 503 cos (x)

66 sin (x) - 50 (3 cos (x) + sin (x))

乘开

- 50 (3 cos (x) + sin (x)) : - 503 cos (x) - 50 sin (x)

- 50 (3 cos (x) + sin (x))

使用分配律:

a (b + c) = ab + a c

a = - 50, b = 3 cos (x), c = sin (x)

= - 503 cos (x) + (- 50) sin (x)

使用加减运算法则

+ (- a) = - a

= - 503 cos (x) - 50 sin (x)

= 66 sin (x) - 503 cos (x) - 50 sin (x)

同类项相加:

66 sin (x) - 50 sin (x) = 16 sin (x)

= 16 sin (x) - 503 cos (x)

= \frac{16 sin ( x ) - 50 3 cos ( x )}{33 ( 3 cos ( x ) + sin ( x ) )}

\frac{16 sin ( x ) - 50 3 cos ( x )}{33 ( 3 cos ( x ) + sin ( x ) )} = 0

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

16 sin (x) - 503 cos (x) = 0

使用三角恒等式改写

16 sin (x) - 503 cos (x) = 0

在两边除以

cos (x), cos (x) \neq = 0

\frac{16 sin ( x ) - 50 3 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{16 sin ( x )}{cos ( x )} - 503 = 0

使用基本三角恒等式:

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

16 tan (x) - 503 = 0

16 tan (x) - 503 = 0

将

503

到右边

16 tan (x) - 503 = 0

两边加上

503

16 tan (x) - 503 + 503 = 0 + 503

化简

16 tan (x) = 503

16 tan (x) = 503

两边除以

16

16 tan (x) = 503

两边除以

16

\frac{16 tan ( x )}{16} = \frac{50 3}{16}

化简

tan (x) = \frac{25 3}{8}

tan (x) = \frac{25 3}{8}

使用反三角函数性质

tan (x) = \frac{25 3}{8}

tan (x) = \frac{25 3}{8}

的通解

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

x = arctan (\frac{25 3}{8}) + 18 0^{\circ} n

x = arctan (\frac{25 3}{8}) + 18 0^{\circ} n

以小数形式表示解

x = 1.38810 \dots + 18 0^{\circ} n

作图

流行的例子

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cot (x) csc (x) = cos (x)

2tan(x)=sqrt(2)

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0.5 = cos (\frac{π}{6} x)

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cot^{2} (θ) + csc (θ) = 1, 0 \leq θ < 2 π