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受欢迎的三角函数 >

3tan(x+43)=2cos(x+43)

解答

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

解答

x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

+1

弧度

x = - \frac{13 π}{180} + 2 πn, x = \frac{107 π}{180} + 2 πn

求解步骤

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

两边减去

2 cos (x + 4 3^{\circ})

3 tan (x + 4 3^{\circ}) - 2 cos (x + 4 3^{\circ}) = 0

化简

3 tan (x + 4 3^{\circ}) - 2 cos (x + 4 3^{\circ}) : 3 tan (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180})

3 tan (x + 4 3^{\circ}) - 2 cos (x + 4 3^{\circ})

化简

x + 4 3^{\circ} : \frac{180 x + 774 0 ^{\circ}}{180}

x + 4 3^{\circ}

将项转换为分式:

x = \frac{x 180}{180}

= \frac{x \cdot 180}{180} + 4 3^{\circ}

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{x \cdot 180 + 774 0 ^{\circ}}{180}

= 3 tan (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos (x + 4 3^{\circ})

化简

x + 4 3^{\circ} : \frac{180 x + 774 0 ^{\circ}}{180}

x + 4 3^{\circ}

将项转换为分式:

x = \frac{x 180}{180}

= \frac{x \cdot 180}{180} + 4 3^{\circ}

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{x \cdot 180 + 774 0 ^{\circ}}{180}

= 3 tan (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180})

3 tan (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

用 sin, cos 表示

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

化简

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) : \frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) - 2 cos ^{2} ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180})

乘

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} : \frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

3 \cdot \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

分式相乘:

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

= \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) \cdot 3}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

= \frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180})

将项转换为分式:

2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{2 cos ( \frac{180 x + 774 0 ^{\circ}}{180} ) cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

= \frac{sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) \cdot 3}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} - \frac{2 cos ( \frac{180 x + 774 0 ^{\circ}}{180} ) cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

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{sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) \cdot 3 - 2 cos ( \frac{180 x + 774 0 ^{\circ}}{180} ) cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

sin (\frac{180 x + 774 0 ^{\circ}}{180}) \cdot 3 - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180}) = 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

sin (\frac{180 x + 774 0 ^{\circ}}{180}) \cdot 3 - 2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180})

2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180}) = 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

2 cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180})

使用指数法则:

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

cos (\frac{180 x + 774 0 ^{\circ}}{180}) cos (\frac{180 x + 774 0 ^{\circ}}{180}) = cos^{1 + 1} (\frac{180 x + 774 0 ^{\circ}}{180})

= 2 cos^{1 + 1} (\frac{180 x + 774 0 ^{\circ}}{180})

数字相加:

1 + 1 = 2

= 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

= 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

= \frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) - 2 cos ^{2} ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )}

\frac{3 sin ( \frac{180 x + 774 0 ^{\circ}}{180} ) - 2 cos ^{2} ( \frac{180 x + 774 0 ^{\circ}}{180} )}{cos ( \frac{180 x + 774 0 ^{\circ}}{180} )} = 0

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

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

两边加上

2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})

两边进行平方

(3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} = (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

两边减去

(2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

分解

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180}) : (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

将

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

改写为

(3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

9 sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

将

9

改写为

3^{2}

= 3^{2} sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 4 cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

将

4

改写为

2^{2}

= 3^{2} sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 2^{2} cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180})

使用指数法则:

a^{b c} = (a^{b})^{c}

cos^{4} (\frac{180 x + 774 0 ^{\circ}}{180}) = (cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

= 3^{2} sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) - 2^{2} (cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

使用指数法则:

a^{m} b^{m} = (ab)^{m}

3^{2} sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

= (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - 2^{2} (cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

使用指数法则:

a^{m} b^{m} = (ab)^{m}

2^{2} (cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} = (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

= (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

= (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

(3 sin (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} - (2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))^{2} = (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))

= (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}))

(3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) (3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) = 0

分别求解每个部分

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0 or 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0 : x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) + 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

使用三角恒等式改写

2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180})

使用毕达哥拉斯恒等式:

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

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

= 2 (1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180})

(1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) \cdot 2 + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

用替代法求解

(1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) \cdot 2 + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

令

: sin (\frac{180 x + 774 0 ^{\circ}}{180}) = u

(1 - u^{2}) \cdot 2 + 3 u = 0

(1 - u^{2}) \cdot 2 + 3 u = 0 : u = - \frac{1}{2}, u = 2

(1 - u^{2}) \cdot 2 + 3 u = 0

展开

(1 - u^{2}) \cdot 2 + 3 u : 2 - 2 u^{2} + 3 u

(1 - u^{2}) \cdot 2 + 3 u

= 2 (1 - u^{2}) + 3 u

乘开

2 (1 - u^{2}) : 2 - 2 u^{2}

2 (1 - u^{2})

使用分配律:

a (b - c) = ab - a c

a = 2, b = 1, c = u^{2}

= 2 \cdot 1 - 2 u^{2}

数字相乘:

2 \cdot 1 = 2

= 2 - 2 u^{2}

= 2 - 2 u^{2} + 3 u

2 - 2 u^{2} + 3 u = 0

改写成标准形式

a x^{2} + b x + c = 0

- 2 u^{2} + 3 u + 2 = 0

使用求根公式求解

- 2 u^{2} + 3 u + 2 = 0

二次方程求根公式:

若

a = - 2, b = 3, c = 2

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 2 ) \cdot 2}{2 ( - 2 )}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 ( - 2 ) \cdot 2}{2 ( - 2 )}

3^{2} - 4 (- 2) \cdot 2 = 5

3^{2} - 4 (- 2) \cdot 2

使用法则

- (- a) = a

= 3^{2} + 4 \cdot 2 \cdot 2

数字相乘:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数字相加:

9 + 16 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 ( - 2 )}

将解分隔开

u_{1} = \frac{- 3 + 5}{2 ( - 2 )}, u_{2} = \frac{- 3 - 5}{2 ( - 2 )}

u = \frac{- 3 + 5}{2 ( - 2 )} : - \frac{1}{2}

\frac{- 3 + 5}{2 ( - 2 )}

去除括号:

(- a) = - a

= \frac{- 3 + 5}{- 2 \cdot 2}

数字相加/相减:

- 3 + 5 = 2

= \frac{2}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{2}{- 4}

使用分式法则:

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

= - \frac{2}{4}

约分:

2

= - \frac{1}{2}

u = \frac{- 3 - 5}{2 ( - 2 )} : 2

\frac{- 3 - 5}{2 ( - 2 )}

去除括号:

(- a) = - a

= \frac{- 3 - 5}{- 2 \cdot 2}

数字相减:

- 3 - 5 = - 8

= \frac{- 8}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 8}{- 4}

使用分式法则:

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

= \frac{8}{4}

数字相除:

\frac{8}{4} = 2

= 2

二次方程组的解是:

u = - \frac{1}{2}, u = 2

u = sin (\frac{180 x + 774 0 ^{\circ}}{180})

代回

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2}, sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2}, sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2} : x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2}

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - \frac{1}{2}

的通解

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{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n, \frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n, \frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n

解

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n : x = 16 7^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n

在两边乘以

180

\frac{180 x + 774 0 ^{\circ}}{180} = 21 0^{\circ} + 36 0^{\circ} n

在两边乘以

180

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 21 0^{\circ} + 180 \cdot 36 0^{\circ} n

化简

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 21 0^{\circ} + 180 \cdot 36 0^{\circ} n

化简

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} : 180 x + 774 0^{\circ}

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

数字相除:

\frac{180}{180} = 1

= 180 x + 774 0^{\circ}

化简

180 \cdot 21 0^{\circ} + 180 \cdot 36 0^{\circ} n : 3780 0^{\circ} + 6480 0^{\circ} n

180 \cdot 21 0^{\circ} + 180 \cdot 36 0^{\circ} n

180 \cdot 21 0^{\circ} = 3780 0^{\circ}

180 \cdot 21 0^{\circ}

分式相乘:

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

= 3780 0^{\circ}

数字相乘:

7 \cdot 180 = 1260

= 3780 0^{\circ}

数字相除:

\frac{1260}{6} = 210

= 3780 0^{\circ}

180 \cdot 36 0^{\circ} n = 6480 0^{\circ} n

180 \cdot 36 0^{\circ} n

数字相乘:

180 \cdot 2 = 360

= 6480 0^{\circ} n

= 3780 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n

将

774 0^{\circ}

到右边

180 x + 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n

两边减去

774 0^{\circ}

180 x + 774 0^{\circ} - 774 0^{\circ} = 3780 0^{\circ} + 6480 0^{\circ} n - 774 0^{\circ}

化简

180 x = 3006 0^{\circ} + 6480 0^{\circ} n

180 x = 3006 0^{\circ} + 6480 0^{\circ} n

两边除以

180

180 x = 3006 0^{\circ} + 6480 0^{\circ} n

两边除以

180

\frac{180 x}{180} = 16 7^{\circ} + \frac{6480 0 ^{\circ} n}{180}

化简

x = 16 7^{\circ} + 36 0^{\circ} n

x = 16 7^{\circ} + 36 0^{\circ} n

解

\frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n : x = 28 7^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n

在两边乘以

180

\frac{180 x + 774 0 ^{\circ}}{180} = 33 0^{\circ} + 36 0^{\circ} n

在两边乘以

180

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 33 0^{\circ} + 180 \cdot 36 0^{\circ} n

化简

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 33 0^{\circ} + 180 \cdot 36 0^{\circ} n

化简

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} : 180 x + 774 0^{\circ}

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

数字相除:

\frac{180}{180} = 1

= 180 x + 774 0^{\circ}

化简

180 \cdot 33 0^{\circ} + 180 \cdot 36 0^{\circ} n : 5940 0^{\circ} + 6480 0^{\circ} n

180 \cdot 33 0^{\circ} + 180 \cdot 36 0^{\circ} n

180 \cdot 33 0^{\circ} = 5940 0^{\circ}

180 \cdot 33 0^{\circ}

分式相乘:

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

= 5940 0^{\circ}

数字相乘:

11 \cdot 180 = 1980

= 5940 0^{\circ}

数字相除:

\frac{1980}{6} = 330

= 5940 0^{\circ}

180 \cdot 36 0^{\circ} n = 6480 0^{\circ} n

180 \cdot 36 0^{\circ} n

数字相乘:

180 \cdot 2 = 360

= 6480 0^{\circ} n

= 5940 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n

将

774 0^{\circ}

到右边

180 x + 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n

两边减去

774 0^{\circ}

180 x + 774 0^{\circ} - 774 0^{\circ} = 5940 0^{\circ} + 6480 0^{\circ} n - 774 0^{\circ}

化简

180 x = 5166 0^{\circ} + 6480 0^{\circ} n

180 x = 5166 0^{\circ} + 6480 0^{\circ} n

两边除以

180

180 x = 5166 0^{\circ} + 6480 0^{\circ} n

两边除以

180

\frac{180 x}{180} = 28 7^{\circ} + \frac{6480 0 ^{\circ} n}{180}

化简

x = 28 7^{\circ} + 36 0^{\circ} n

x = 28 7^{\circ} + 36 0^{\circ} n

x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2 :

无解

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 2

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0 : x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) - 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

使用三角恒等式改写

- 2 cos^{2} (\frac{180 x + 774 0 ^{\circ}}{180}) + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180})

使用毕达哥拉斯恒等式:

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

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

= - 2 (1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180})

- (1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) \cdot 2 + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

用替代法求解

- (1 - sin^{2} (\frac{180 x + 774 0 ^{\circ}}{180})) \cdot 2 + 3 sin (\frac{180 x + 774 0 ^{\circ}}{180}) = 0

令

: sin (\frac{180 x + 774 0 ^{\circ}}{180}) = u

- (1 - u^{2}) \cdot 2 + 3 u = 0

- (1 - u^{2}) \cdot 2 + 3 u = 0 : u = \frac{1}{2}, u = - 2

- (1 - u^{2}) \cdot 2 + 3 u = 0

展开

- (1 - u^{2}) \cdot 2 + 3 u : - 2 + 2 u^{2} + 3 u

- (1 - u^{2}) \cdot 2 + 3 u

= - 2 (1 - u^{2}) + 3 u

乘开

- 2 (1 - u^{2}) : - 2 + 2 u^{2}

- 2 (1 - u^{2})

使用分配律:

a (b - c) = ab - a c

a = - 2, b = 1, c = u^{2}

= - 2 \cdot 1 - (- 2) u^{2}

使用加减运算法则

- (- a) = a

= - 2 \cdot 1 + 2 u^{2}

数字相乘:

2 \cdot 1 = 2

= - 2 + 2 u^{2}

= - 2 + 2 u^{2} + 3 u

- 2 + 2 u^{2} + 3 u = 0

改写成标准形式

a x^{2} + b x + c = 0

2 u^{2} + 3 u - 2 = 0

使用求根公式求解

2 u^{2} + 3 u - 2 = 0

二次方程求根公式:

若

a = 2, b = 3, c = - 2

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

3^{2} - 4 \cdot 2 (- 2) = 5

3^{2} - 4 \cdot 2 (- 2)

使用法则

- (- a) = a

= 3^{2} + 4 \cdot 2 \cdot 2

数字相乘:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数字相加:

9 + 16 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- 3 + 5}{2 \cdot 2}, u_{2} = \frac{- 3 - 5}{2 \cdot 2}

u = \frac{- 3 + 5}{2 \cdot 2} : \frac{1}{2}

\frac{- 3 + 5}{2 \cdot 2}

数字相加/相减:

- 3 + 5 = 2

= \frac{2}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{2}{4}

约分:

2

= \frac{1}{2}

u = \frac{- 3 - 5}{2 \cdot 2} : - 2

\frac{- 3 - 5}{2 \cdot 2}

数字相减:

- 3 - 5 = - 8

= \frac{- 8}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 8}{4}

使用分式法则:

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

= - \frac{8}{4}

数字相除:

\frac{8}{4} = 2

= - 2

二次方程组的解是:

u = \frac{1}{2}, u = - 2

u = sin (\frac{180 x + 774 0 ^{\circ}}{180})

代回

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2}, sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - 2

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2}, sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - 2

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2} : x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2}

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = \frac{1}{2}

的通解

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{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n, \frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n, \frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n

解

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n : x = - 1 3^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n

在两边乘以

180

\frac{180 x + 774 0 ^{\circ}}{180} = 3 0^{\circ} + 36 0^{\circ} n

在两边乘以

180

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 3 0^{\circ} + 180 \cdot 36 0^{\circ} n

化简

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 3 0^{\circ} + 180 \cdot 36 0^{\circ} n

化简

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} : 180 x + 774 0^{\circ}

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

数字相除:

\frac{180}{180} = 1

= 180 x + 774 0^{\circ}

化简

180 \cdot 3 0^{\circ} + 180 \cdot 36 0^{\circ} n : 540 0^{\circ} + 6480 0^{\circ} n

180 \cdot 3 0^{\circ} + 180 \cdot 36 0^{\circ} n

180 \cdot 3 0^{\circ} = 540 0^{\circ}

180 \cdot 3 0^{\circ}

分式相乘:

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

= 540 0^{\circ}

数字相除:

\frac{180}{6} = 30

= 540 0^{\circ}

180 \cdot 36 0^{\circ} n = 6480 0^{\circ} n

180 \cdot 36 0^{\circ} n

数字相乘:

180 \cdot 2 = 360

= 6480 0^{\circ} n

= 540 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n

将

774 0^{\circ}

到右边

180 x + 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n

两边减去

774 0^{\circ}

180 x + 774 0^{\circ} - 774 0^{\circ} = 540 0^{\circ} + 6480 0^{\circ} n - 774 0^{\circ}

化简

180 x = - 234 0^{\circ} + 6480 0^{\circ} n

180 x = - 234 0^{\circ} + 6480 0^{\circ} n

两边除以

180

180 x = - 234 0^{\circ} + 6480 0^{\circ} n

两边除以

180

\frac{180 x}{180} = - 1 3^{\circ} + \frac{6480 0 ^{\circ} n}{180}

化简

x = - 1 3^{\circ} + 36 0^{\circ} n

x = - 1 3^{\circ} + 36 0^{\circ} n

解

\frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n : x = 10 7^{\circ} + 36 0^{\circ} n

\frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n

在两边乘以

180

\frac{180 x + 774 0 ^{\circ}}{180} = 15 0^{\circ} + 36 0^{\circ} n

在两边乘以

180

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 15 0^{\circ} + 180 \cdot 36 0^{\circ} n

化简

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} = 180 \cdot 15 0^{\circ} + 180 \cdot 36 0^{\circ} n

化简

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180} : 180 x + 774 0^{\circ}

\frac{180 ( 180 x + 774 0 ^{\circ} )}{180}

数字相除:

\frac{180}{180} = 1

= 180 x + 774 0^{\circ}

化简

180 \cdot 15 0^{\circ} + 180 \cdot 36 0^{\circ} n : 2700 0^{\circ} + 6480 0^{\circ} n

180 \cdot 15 0^{\circ} + 180 \cdot 36 0^{\circ} n

180 \cdot 15 0^{\circ} = 2700 0^{\circ}

180 \cdot 15 0^{\circ}

分式相乘:

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

= 2700 0^{\circ}

数字相乘:

5 \cdot 180 = 900

= 2700 0^{\circ}

数字相除:

\frac{900}{6} = 150

= 2700 0^{\circ}

180 \cdot 36 0^{\circ} n = 6480 0^{\circ} n

180 \cdot 36 0^{\circ} n

数字相乘:

180 \cdot 2 = 360

= 6480 0^{\circ} n

= 2700 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n

180 x + 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n

将

774 0^{\circ}

到右边

180 x + 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n

两边减去

774 0^{\circ}

180 x + 774 0^{\circ} - 774 0^{\circ} = 2700 0^{\circ} + 6480 0^{\circ} n - 774 0^{\circ}

化简

180 x = 1926 0^{\circ} + 6480 0^{\circ} n

180 x = 1926 0^{\circ} + 6480 0^{\circ} n

两边除以

180

180 x = 1926 0^{\circ} + 6480 0^{\circ} n

两边除以

180

\frac{180 x}{180} = 10 7^{\circ} + \frac{6480 0 ^{\circ} n}{180}

化简

x = 10 7^{\circ} + 36 0^{\circ} n

x = 10 7^{\circ} + 36 0^{\circ} n

x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - 2 :

无解

sin (\frac{180 x + 774 0 ^{\circ}}{180}) = - 2

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

合并所有解

x = 16 7^{\circ} + 36 0^{\circ} n, x = 28 7^{\circ} + 36 0^{\circ} n, x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

将解代入原方程进行验证

将它们代入

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

检验解是否符合

去除与方程不符的解。

检验

16 7^{\circ} + 36 0^{\circ} n

的解:

假

16 7^{\circ} + 36 0^{\circ} n

代入

n = 1

16 7^{\circ} + 36 0^{\circ} 1

对于

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

代入

x = 16 7^{\circ} + 36 0^{\circ} 1

3 tan (16 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ}) = 2 cos (16 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ})

整理后得

1.73205 \dots = - 1.73205 \dots

\Rightarrow 假

检验

28 7^{\circ} + 36 0^{\circ} n

的解:

假

28 7^{\circ} + 36 0^{\circ} n

代入

n = 1

28 7^{\circ} + 36 0^{\circ} 1

对于

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

代入

x = 28 7^{\circ} + 36 0^{\circ} 1

3 tan (28 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ}) = 2 cos (28 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ})

整理后得

- 1.73205 \dots = 1.73205 \dots

\Rightarrow 假

检验

- 1 3^{\circ} + 36 0^{\circ} n

的解:

真

- 1 3^{\circ} + 36 0^{\circ} n

代入

n = 1

- 1 3^{\circ} + 36 0^{\circ} 1

对于

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

代入

x = - 1 3^{\circ} + 36 0^{\circ} 1

3 tan (- 1 3^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ}) = 2 cos (- 1 3^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ})

整理后得

1.73205 \dots = 1.73205 \dots

\Rightarrow 真

检验

10 7^{\circ} + 36 0^{\circ} n

的解:

真

10 7^{\circ} + 36 0^{\circ} n

代入

n = 1

10 7^{\circ} + 36 0^{\circ} 1

对于

3 tan (x + 4 3^{\circ}) = 2 cos (x + 4 3^{\circ})

代入

x = 10 7^{\circ} + 36 0^{\circ} 1

3 tan (10 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ}) = 2 cos (10 7^{\circ} + 36 0^{\circ} 1 + 4 3^{\circ})

整理后得

- 1.73205 \dots = - 1.73205 \dots

\Rightarrow 真

x = - 1 3^{\circ} + 36 0^{\circ} n, x = 10 7^{\circ} + 36 0^{\circ} n

作图

流行的例子

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4 sin (x) = 3 sec (x)

(sin(55.01-x))/(sin(x))= 1200/500

\frac{sin ( 55.01 - x )}{sin ( x )} = \frac{1200}{500}

csc (θ) = 0.77

(sin(x)+cos(x))/(sin(x))=1 1/(tan(x))

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

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arctan (x) = \frac{- π}{2}