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受欢迎的三角函数 >

3sin(x)=2sec(x)tan(x)

解答

3 sin (x) = 2 sec (x) tan (x)

解答

x = 2 πn, x = π + 2 πn, x = 2.52611 \dots + 2 πn, x = - 2.52611 \dots + 2 πn, x = 0.61547 \dots + 2 πn, x = 2 π - 0.61547 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 144.73561 \dots^{\circ} + 36 0^{\circ} n, x = - 144.73561 \dots^{\circ} + 36 0^{\circ} n, x = 35.26438 \dots^{\circ} + 36 0^{\circ} n, x = 324.73561 \dots^{\circ} + 36 0^{\circ} n

求解步骤

3 sin (x) = 2 sec (x) tan (x)

两边减去

2 sec (x) tan (x)

3 sin (x) - 2 sec (x) tan (x) = 0

用 sin, cos 表示

3 sin (x) - 2 sec (x) tan (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

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

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

分式相乘:

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

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

数字相乘:

1 \cdot 2 = 2

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

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

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)

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

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

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

将项转换为分式:

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

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

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

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

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

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

分解

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

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

因式分解出通项

sin (x)

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

分解

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

3 cos^{2} (x) - 2

将

3 cos^{2} (x) - 2

改写为

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

3 cos^{2} (x) - 2

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

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

使用根式运算法则:

a = (a)^{2}

2 = (2)^{2}

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

使用指数法则:

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

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

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

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

使用平方差公式:

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

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

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

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

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

分别求解每个部分

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0

的通解

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

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

3 cos (x) + 2 = 0

将

2

到右边

3 cos (x) + 2 = 0

两边减去

2

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

化简

3 cos (x) = - 2

3 cos (x) = - 2

两边除以

3

3 cos (x) = - 2

两边除以

3

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

化简

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

化简

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

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

约分:

3

= cos (x)

化简

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

\frac{- 2}{3}

使用分式法则:

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

= - \frac{2}{3}

合并相同指数项 :

\frac{x}{y} = \frac{x}{y}

= - \frac{2}{3}

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

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

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

使用反三角函数性质

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

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

的通解

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

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

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

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

3 cos (x) - 2 = 0

将

2

到右边

3 cos (x) - 2 = 0

两边加上

2

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

化简

3 cos (x) = 2

3 cos (x) = 2

两边除以

3

3 cos (x) = 2

两边除以

3

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

化简

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

化简

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

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

约分:

3

= cos (x)

化简

\frac{2}{3} : \frac{2}{3}

\frac{2}{3}

合并相同指数项 :

\frac{x}{y} = \frac{x}{y}

= \frac{2}{3}

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

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

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

使用反三角函数性质

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

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

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

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

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

合并所有解

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

以小数形式表示解

x = 2 πn, x = π + 2 πn, x = 2.52611 \dots + 2 πn, x = - 2.52611 \dots + 2 πn, x = 0.61547 \dots + 2 πn, x = 2 π - 0.61547 \dots + 2 πn

作图

流行的例子

12sin(x)+5cos(x)=0

12 sin (x) + 5 cos (x) = 0

sin(x)=2sin(2x)

sin (x) = 2 sin (2 x)

9sin(x-0.5pi)+8=0

9 sin (x - 0.5 π) + 8 = 0

sec(-135)-cot(x)=2

sec (- 13 5^{\circ}) - cot (x) = 2

csc (x - 3 0^{\circ}) = 2