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受欢迎的三角函数 >

cos^2(x)= 3/(4*5cos^2(x))

解答

cos^{2} (x) = \frac{3}{4 \cdot 5 cos ^{2} ( x )}

解答

x = 0.89907 \dots + 2 πn, x = 2 π - 0.89907 \dots + 2 πn, x = 2.24251 \dots + 2 πn, x = - 2.24251 \dots + 2 πn

+1

度数

x = 51.51329 \dots^{\circ} + 36 0^{\circ} n, x = 308.48670 \dots^{\circ} + 36 0^{\circ} n, x = 128.48670 \dots^{\circ} + 36 0^{\circ} n, x = - 128.48670 \dots^{\circ} + 36 0^{\circ} n

求解步骤

cos^{2} (x) = \frac{3}{4 \cdot 5 cos ^{2} ( x )}

用替代法求解

cos^{2} (x) = \frac{3}{4 \cdot 5 cos ^{2} ( x )}

令

: cos (x) = u

u^{2} = \frac{3}{4 \cdot 5 u ^{2}}

u^{2} = \frac{3}{4 \cdot 5 u ^{2}} : u = \frac{15}{10}, u = - \frac{15}{10}, u = i \frac{15}{10}, u = - i \frac{15}{10}

u^{2} = \frac{3}{4 \cdot 5 u ^{2}}

化简

\frac{3}{4 \cdot 5 u ^{2}} : \frac{3}{20 u ^{2}}

\frac{3}{4 \cdot 5 u ^{2}}

数字相乘:

4 \cdot 5 = 20

= \frac{3}{20 u ^{2}}

u^{2} = \frac{3}{20 u ^{2}}

在两边乘以

u^{2}

u^{2} = \frac{3}{20 u ^{2}}

在两边乘以

u^{2}

u^{2} u^{2} = \frac{3}{20 u ^{2}} u^{2}

化简

u^{2} u^{2} : u^{4}

u^{2} u^{2} = \frac{3}{20 u ^{2}} u^{2}

使用指数法则:

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

u^{2} u^{2} = u^{2 + 2}

= u^{2 + 2}

数字相加:

2 + 2 = 4

= u^{4}

u^{4} = \frac{3}{20}

u^{4} = \frac{3}{20}

解

u^{4} = \frac{3}{20} : u = \frac{15}{10}, u = - \frac{15}{10}, u = i \frac{15}{10}, u = - i \frac{15}{10}

u^{4} = \frac{3}{20}

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

v^{2} = \frac{3}{20}

解

v^{2} = \frac{3}{20} : v = \frac{3}{20}, v = - \frac{3}{20}

v^{2} = \frac{3}{20}

对于

(g (x))^{2} = f (a)

解为

g (x) = f (a), - f (a)

v = \frac{3}{20}, v = - \frac{3}{20}

v = \frac{3}{20}, v = - \frac{3}{20}

代回

v = u^{2}

，求解

u

解

u^{2} = \frac{3}{20} : u = \frac{15}{10}, u = - \frac{15}{10}

u^{2} = \frac{3}{20}

化简

\frac{3}{20} : \frac{15}{10}

\frac{3}{20}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3}{20}

20 = 25

20

20

质因数分解:

2^{2} \cdot 5

20

20

除以

220 = 10 \cdot 2

= 2 \cdot 10

10

除以

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 5

2, 5

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 5

= 2^{2} \cdot 5

= 2^{2} \cdot 5

使用根式运算法则:

n ab = n a n b

= 5 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 25

= \frac{3}{2 5}

\frac{3}{2 5}

有理化:

\frac{15}{10}

\frac{3}{2 5}

乘以共轭根式

\frac{5}{5}

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

35 = 15

35

使用根式运算法则:

a b = a \cdot b

35 = 3 \cdot 5

= 3 \cdot 5

数字相乘:

3 \cdot 5 = 15

= 15

255 = 10

255

使用根式运算法则:

a a = a

55 = 5

= 2 \cdot 5

数字相乘:

2 \cdot 5 = 10

= 10

= \frac{15}{10}

= \frac{15}{10}

u^{2} = \frac{15}{10}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{15}{10}, u = - \frac{15}{10}

解

u^{2} = - \frac{3}{20} : u = i \frac{15}{10}, u = - i \frac{15}{10}

u^{2} = - \frac{3}{20}

化简

- \frac{3}{20} : - \frac{15}{10}

- \frac{3}{20}

化简

\frac{3}{20} : \frac{3}{2 5}

\frac{3}{20}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3}{20}

20 = 25

20

20

质因数分解:

2^{2} \cdot 5

20

20

除以

220 = 10 \cdot 2

= 2 \cdot 10

10

除以

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 5

2, 5

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 5

= 2^{2} \cdot 5

= 2^{2} \cdot 5

使用根式运算法则:

n ab = n a n b

= 5 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 25

= \frac{3}{2 5}

= - \frac{3}{2 5}

- \frac{3}{2 5}

有理化:

- \frac{15}{10}

- \frac{3}{2 5}

乘以共轭根式

\frac{5}{5}

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

35 = 15

35

使用根式运算法则:

a b = a \cdot b

35 = 3 \cdot 5

= 3 \cdot 5

数字相乘:

3 \cdot 5 = 15

= 15

255 = 10

255

使用根式运算法则:

a a = a

55 = 5

= 2 \cdot 5

数字相乘:

2 \cdot 5 = 10

= 10

= - \frac{15}{10}

= - \frac{15}{10}

u^{2} = - \frac{15}{10}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = - \frac{15}{10}, u = - - \frac{15}{10}

化简

- \frac{15}{10} : i \frac{15}{10}

- \frac{15}{10}

使用根式运算法则:

- a = - 1 a

- \frac{15}{10} = - 1 \frac{15}{10}

= - 1 \frac{15}{10}

使用虚数运算法则:

- 1 = i

= i \frac{15}{10}

将

i \frac{15}{10}

改写成标准复数形式:

\frac{15}{10} i

i \frac{15}{10}

\frac{15}{10} = \frac{3}{2 5}

\frac{15}{10}

\frac{15}{10} = \frac{3}{2 5}

\frac{15}{10}

分解

15 : 35

因式分解

15 = 3 \cdot 5

= 3 \cdot 5

使用根式运算法则:

n ab = n a n b

= 35

分解

10 : 2 \cdot 5

因式分解

10 = 2 \cdot 5

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

消掉

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

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

使用根式运算法则:

n a = a^{\frac{1}{n}}

5 = 5^{\frac{1}{2}}

= \frac{3 \cdot 5 ^{\frac{1}{2}}}{2 \cdot 5}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{5 ^{\frac{1}{2}}}{5 ^{1}} = \frac{1}{5 ^{1 - \frac{1}{2}}}

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

数字相减:

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

= \frac{3}{2 \cdot 5 ^{\frac{1}{2}}}

使用根式运算法则:

a^{\frac{1}{n}} = n a

5^{\frac{1}{2}} = 5

= \frac{3}{2 5}

= \frac{3}{2 5}

= \frac{3}{2 5}

= i \frac{3}{2 5}

\frac{3}{2 5} = \frac{15}{10}

\frac{3}{2 5}

\frac{3}{2 5} = \frac{15}{10}

\frac{3}{2 5}

乘以共轭根式

\frac{5}{5}

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

35 = 15

35

使用根式运算法则:

a b = a \cdot b

35 = 3 \cdot 5

= 3 \cdot 5

数字相乘:

3 \cdot 5 = 15

= 15

255 = 10

255

使用根式运算法则:

a a = a

55 = 5

= 2 \cdot 5

数字相乘:

2 \cdot 5 = 10

= 10

= \frac{15}{10}

= \frac{15}{10}

= \frac{15}{10} i

= \frac{15}{10} i

化简

- - \frac{15}{10} : - i \frac{15}{10}

- - \frac{15}{10}

化简

- \frac{15}{10} : i \frac{15}{10}

- \frac{15}{10}

使用根式运算法则:

- a = - 1 a

- \frac{15}{10} = - 1 \frac{15}{10}

= - 1 \frac{15}{10}

使用虚数运算法则:

- 1 = i

= i \frac{15}{10}

= - i \frac{15}{10}

将

- i \frac{15}{10}

改写成标准复数形式:

- \frac{15}{10} i

- i \frac{15}{10}

\frac{15}{10} = \frac{3}{2 5}

\frac{15}{10}

\frac{15}{10} = \frac{3}{2 5}

\frac{15}{10}

分解

15 : 35

因式分解

15 = 3 \cdot 5

= 3 \cdot 5

使用根式运算法则:

n ab = n a n b

= 35

分解

10 : 2 \cdot 5

因式分解

10 = 2 \cdot 5

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

消掉

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

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

使用根式运算法则:

n a = a^{\frac{1}{n}}

5 = 5^{\frac{1}{2}}

= \frac{3 \cdot 5 ^{\frac{1}{2}}}{2 \cdot 5}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{5 ^{\frac{1}{2}}}{5 ^{1}} = \frac{1}{5 ^{1 - \frac{1}{2}}}

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

数字相减:

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

= \frac{3}{2 \cdot 5 ^{\frac{1}{2}}}

使用根式运算法则:

a^{\frac{1}{n}} = n a

5^{\frac{1}{2}} = 5

= \frac{3}{2 5}

= \frac{3}{2 5}

= \frac{3}{2 5}

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

- \frac{3}{2 5} = - \frac{15}{10}

- \frac{3}{2 5}

\frac{3}{2 5} = \frac{15}{10}

\frac{3}{2 5}

乘以共轭根式

\frac{5}{5}

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

35 = 15

35

使用根式运算法则:

a b = a \cdot b

35 = 3 \cdot 5

= 3 \cdot 5

数字相乘:

3 \cdot 5 = 15

= 15

255 = 10

255

使用根式运算法则:

a a = a

55 = 5

= 2 \cdot 5

数字相乘:

2 \cdot 5 = 10

= 10

= \frac{15}{10}

= - \frac{15}{10}

= - \frac{15}{10} i

= - \frac{15}{10} i

u = i \frac{15}{10}, u = - i \frac{15}{10}

解为

u = \frac{15}{10}, u = - \frac{15}{10}, u = i \frac{15}{10}, u = - i \frac{15}{10}

u = \frac{15}{10}, u = - \frac{15}{10}, u = i \frac{15}{10}, u = - i \frac{15}{10}

验证解

找到无定义的点（奇点）:

u = 0

取

\frac{3}{45 u ^{2}}

的分母，令其等于零

解

45 u^{2} = 0 : u = 0

4 \cdot 5 u^{2} = 0

两边除以

20

4 \cdot 5 u^{2} = 0

两边除以

20

4 \cdot 5 u^{2} = 0

两边除以

20

\frac{4 \cdot 5 u ^{2}}{20} = \frac{0}{20}

化简

u^{2} = 0

u^{2} = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = \frac{15}{10}, u = - \frac{15}{10}, u = i \frac{15}{10}, u = - i \frac{15}{10}

u = cos (x)

代回

cos (x) = \frac{15}{10}, cos (x) = - \frac{15}{10}, cos (x) = i \frac{15}{10}, cos (x) = - i \frac{15}{10}

cos (x) = \frac{15}{10}, cos (x) = - \frac{15}{10}, cos (x) = i \frac{15}{10}, cos (x) = - i \frac{15}{10}

cos (x) = \frac{15}{10} : x = arccos \frac{15}{10} + 2 πn, x = 2 π - arccos \frac{15}{10} + 2 πn

cos (x) = \frac{15}{10}

使用反三角函数性质

cos (x) = \frac{15}{10}

cos (x) = \frac{15}{10}

的通解

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

x = arccos \frac{15}{10} + 2 πn, x = 2 π - arccos \frac{15}{10} + 2 πn

x = arccos \frac{15}{10} + 2 πn, x = 2 π - arccos \frac{15}{10} + 2 πn

cos (x) = - \frac{15}{10} : x = arccos - \frac{15}{10} + 2 πn, x = - arccos - \frac{15}{10} + 2 πn

cos (x) = - \frac{15}{10}

使用反三角函数性质

cos (x) = - \frac{15}{10}

cos (x) = - \frac{15}{10}

的通解

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

x = arccos - \frac{15}{10} + 2 πn, x = - arccos - \frac{15}{10} + 2 πn

x = arccos - \frac{15}{10} + 2 πn, x = - arccos - \frac{15}{10} + 2 πn

cos (x) = i \frac{15}{10} :

无解

cos (x) = i \frac{15}{10}

无解

cos (x) = - i \frac{15}{10} :

无解

cos (x) = - i \frac{15}{10}

无解

合并所有解

x = arccos \frac{15}{10} + 2 πn, x = 2 π - arccos \frac{15}{10} + 2 πn, x = arccos - \frac{15}{10} + 2 πn, x = - arccos - \frac{15}{10} + 2 πn

以小数形式表示解

x = 0.89907 \dots + 2 πn, x = 2 π - 0.89907 \dots + 2 πn, x = 2.24251 \dots + 2 πn, x = - 2.24251 \dots + 2 πn

作图

流行的例子

(cos^2(x))/(1-cos(x))=1+cos(x)

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

sin (x) = \frac{19}{50}

5sin(2x-pi/2)=0.5

5 sin (2 x - \frac{π}{2}) = 0.5

5sin(x)=3sin(x)+cos(x)

5 sin (x) = 3 sin (x) + cos (x)

3-3cos(5x)=3cos(5x)

3 - 3 cos (5 x) = 3 cos (5 x)