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受欢迎的三角函数 >

1/(cos^2(x))= 2/(3-3sin(x))

解答

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

解答

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度数

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

求解步骤

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

两边减去

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

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

化简

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

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

分解

3 - 3 sin (x) : - 3 (sin (x) - 1)

3 - 3 sin (x)

因式分解出通项

- 3

= - 3 (sin (x) - 1)

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

cos^{2} (x), - 3 (sin (x) - 1)

的最小公倍数:

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

cos^{2} (x), - 3 (sin (x) - 1)

最小公倍数 (LCM)

计算出由出现在

cos^{2} (x)

或

- 3 (sin (x) - 1)

中的因子组成的表达式

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

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

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

对于

\frac{1}{cos ^{2} ( x )} :

将分母和分子乘以

- 3 (sin (x) - 1)

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

对于

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

将分母和分子乘以

cos^{2} (x)

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

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

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 ) - 1 ) - 2 cos ^{2} ( x )}{- 3 cos ^{2} ( x ) ( sin ( x ) - 1 )}

使用分式法则:

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

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

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

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

- (- 3 (sin (x) - 1) - 2 cos^{2} (x)) = 0

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

= - (- (- 1 + sin (x)) \cdot 3 - 2 (1 - sin^{2} (x)))

化简

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

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

乘开

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

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

= - 3 (- 1 + sin (x)) - 2 (1 - sin^{2} (x))

乘开

- 3 (- 1 + sin (x)) : 3 - 3 sin (x)

- 3 (- 1 + sin (x))

使用分配律:

a (b + c) = ab + a c

a = - 3, b = - 1, c = sin (x)

= - 3 (- 1) + (- 3) sin (x)

使用加减运算法则

- (- a) = a, + (- a) = - a

= 3 \cdot 1 - 3 sin (x)

数字相乘:

3 \cdot 1 = 3

= 3 - 3 sin (x)

= 3 - 3 sin (x) - 2 (1 - sin^{2} (x))

乘开

- 2 (1 - sin^{2} (x)) : - 2 + 2 sin^{2} (x)

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

使用分配律:

a (b - c) = ab - a c

a = - 2, b = 1, c = sin^{2} (x)

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

使用加减运算法则

- (- a) = a

= - 2 \cdot 1 + 2 sin^{2} (x)

数字相乘:

2 \cdot 1 = 2

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

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

化简

3 - 3 sin (x) - 2 + 2 sin^{2} (x) : 2 sin^{2} (x) - 3 sin (x) + 1

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

对同类项分组

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

数字相加/相减:

3 - 2 = 1

= 2 sin^{2} (x) - 3 sin (x) + 1

= 2 sin^{2} (x) - 3 sin (x) + 1

= - (2 sin^{2} (x) - 3 sin (x) + 1)

打开括号

= - (2 sin^{2} (x)) - (- 3 sin (x)) - (1)

使用加减运算法则

- (- a) = a, - (a) = - a

= - 2 sin^{2} (x) + 3 sin (x) - 1

= - 2 sin^{2} (x) + 3 sin (x) - 1

- 1 - 2 sin^{2} (x) + 3 sin (x) = 0

用替代法求解

- 1 - 2 sin^{2} (x) + 3 sin (x) = 0

令

: sin (x) = u

- 1 - 2 u^{2} + 3 u = 0

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

- 1 - 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 - 1 = 0

使用求根公式求解

- 2 u^{2} + 3 u - 1 = 0

二次方程求根公式:

若

a = - 2, b = 3, c = - 1

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

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

3^{2} - 4 (- 2) (- 1) = 1

3^{2} - 4 (- 2) (- 1)

使用法则

- (- a) = a

= 3^{2} - 4 \cdot 2 \cdot 1

数字相乘:

4 \cdot 2 \cdot 1 = 8

= 3^{2} - 8

3^{2} = 9

= 9 - 8

数字相减:

9 - 8 = 1

= 1

使用法则

1 = 1

= 1

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

将解分隔开

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

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

\frac{- 3 + 1}{2 ( - 2 )}

去除括号:

(- a) = - a

= \frac{- 3 + 1}{- 2 \cdot 2}

数字相加/相减:

- 3 + 1 = - 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 - 1}{2 ( - 2 )} : 1

\frac{- 3 - 1}{2 ( - 2 )}

去除括号:

(- a) = - a

= \frac{- 3 - 1}{- 2 \cdot 2}

数字相减:

- 3 - 1 = - 4

= \frac{- 4}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 4}{- 4}

使用分式法则:

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

= \frac{4}{4}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

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

u = sin (x)

代回

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

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

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

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

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 = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

sin (x) = 1

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 = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{π}{2} + 2 πn

因为方程对以下值无定义:

\frac{π}{2} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

作图

流行的例子

cos (4 x) = - 1

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

cos (x) = sec (x)

cos^{2} (x) = 2

2sin^2(x)=3cos(x)

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