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受欢迎的三角函数 >

3sec^2(x)-5=4tan(x)

解答

3 sec^{2} (x) - 5 = 4 tan (x)

解答

x = 1.04436 \dots + πn, x = - 0.36961 \dots + πn

+1

度数

x = 59.83746 \dots^{\circ} + 18 0^{\circ} n, x = - 21.17765 \dots^{\circ} + 18 0^{\circ} n

求解步骤

3 sec^{2} (x) - 5 = 4 tan (x)

两边减去

4 tan (x)

3 sec^{2} (x) - 5 - 4 tan (x) = 0

使用三角恒等式改写

- 5 + 3 sec^{2} (x) - 4 tan (x)

使用毕达哥拉斯恒等式:

sec^{2} (x) = tan^{2} (x) + 1

= - 5 + 3 (tan^{2} (x) + 1) - 4 tan (x)

化简

- 5 + 3 (tan^{2} (x) + 1) - 4 tan (x) : 3 tan^{2} (x) - 4 tan (x) - 2

- 5 + 3 (tan^{2} (x) + 1) - 4 tan (x)

乘开

3 (tan^{2} (x) + 1) : 3 tan^{2} (x) + 3

3 (tan^{2} (x) + 1)

使用分配律:

a (b + c) = ab + a c

a = 3, b = tan^{2} (x), c = 1

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

数字相乘:

3 \cdot 1 = 3

= 3 tan^{2} (x) + 3

= - 5 + 3 tan^{2} (x) + 3 - 4 tan (x)

化简

- 5 + 3 tan^{2} (x) + 3 - 4 tan (x) : 3 tan^{2} (x) - 4 tan (x) - 2

- 5 + 3 tan^{2} (x) + 3 - 4 tan (x)

对同类项分组

= 3 tan^{2} (x) - 4 tan (x) - 5 + 3

数字相加/相减:

- 5 + 3 = - 2

= 3 tan^{2} (x) - 4 tan (x) - 2

= 3 tan^{2} (x) - 4 tan (x) - 2

= 3 tan^{2} (x) - 4 tan (x) - 2

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

用替代法求解

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

令

: tan (x) = u

- 2 + 3 u^{2} - 4 u = 0

- 2 + 3 u^{2} - 4 u = 0 : u = \frac{2 + 10}{3}, u = \frac{2 - 10}{3}

- 2 + 3 u^{2} - 4 u = 0

改写成标准形式

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

3 u^{2} - 4 u - 2 = 0

使用求根公式求解

3 u^{2} - 4 u - 2 = 0

二次方程求根公式:

若

a = 3, b = - 4, c = - 2

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

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

(- 4)^{2} - 4 \cdot 3 (- 2) = 210

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

使用法则

- (- a) = a

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

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

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

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

数字相乘:

4 \cdot 3 \cdot 2 = 24

= 4^{2} + 24

4^{2} = 16

= 16 + 24

数字相加:

16 + 24 = 40

= 40

40

质因数分解:

2^{3} \cdot 5

40

40

除以

240 = 20 \cdot 2

= 2 \cdot 20

20

除以

220 = 10 \cdot 2

= 2 \cdot 2 \cdot 10

10

除以

210 = 5 \cdot 2

= 2 \cdot 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 2 \cdot 5

= 2^{3} \cdot 5

= 2^{3} \cdot 5

使用指数法则:

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

= 2^{2} \cdot 2 \cdot 5

使用根式运算法则:

n ab = n a n b

= 2^{2} 2 \cdot 5

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2 2 \cdot 5

整理后得

= 210

u_{1, 2} = \frac{- ( - 4 ) \pm 2 10}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 4 ) + 2 10}{2 \cdot 3}, u_{2} = \frac{- ( - 4 ) - 2 10}{2 \cdot 3}

u = \frac{- ( - 4 ) + 2 10}{2 \cdot 3} : \frac{2 + 10}{3}

\frac{- ( - 4 ) + 2 10}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{4 + 2 10}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{4 + 2 10}{6}

分解

4 + 210 : 2 (2 + 10)

4 + 210

改写为

= 2 \cdot 2 + 210

因式分解出通项

2

= 2 (2 + 10)

= \frac{2 ( 2 + 10 )}{6}

约分:

2

= \frac{2 + 10}{3}

u = \frac{- ( - 4 ) - 2 10}{2 \cdot 3} : \frac{2 - 10}{3}

\frac{- ( - 4 ) - 2 10}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{4 - 2 10}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{4 - 2 10}{6}

分解

4 - 210 : 2 (2 - 10)

4 - 210

改写为

= 2 \cdot 2 - 210

因式分解出通项

2

= 2 (2 - 10)

= \frac{2 ( 2 - 10 )}{6}

约分:

2

= \frac{2 - 10}{3}

二次方程组的解是:

u = \frac{2 + 10}{3}, u = \frac{2 - 10}{3}

u = tan (x)

代回

tan (x) = \frac{2 + 10}{3}, tan (x) = \frac{2 - 10}{3}

tan (x) = \frac{2 + 10}{3}, tan (x) = \frac{2 - 10}{3}

tan (x) = \frac{2 + 10}{3} : x = arctan (\frac{2 + 10}{3}) + πn

tan (x) = \frac{2 + 10}{3}

使用反三角函数性质

tan (x) = \frac{2 + 10}{3}

tan (x) = \frac{2 + 10}{3}

的通解

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{2 + 10}{3}) + πn

x = arctan (\frac{2 + 10}{3}) + πn

tan (x) = \frac{2 - 10}{3} : x = arctan (\frac{2 - 10}{3}) + πn

tan (x) = \frac{2 - 10}{3}

使用反三角函数性质

tan (x) = \frac{2 - 10}{3}

tan (x) = \frac{2 - 10}{3}

的通解

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (\frac{2 - 10}{3}) + πn

x = arctan (\frac{2 - 10}{3}) + πn

合并所有解

x = arctan (\frac{2 + 10}{3}) + πn, x = arctan (\frac{2 - 10}{3}) + πn

以小数形式表示解

x = 1.04436 \dots + πn, x = - 0.36961 \dots + πn

作图

流行的例子

4tan^2(x)+5tan(x)=6

4 tan^{2} (x) + 5 tan (x) = 6

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

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2 sin^{2} (x) - 5 sin (x) + 3 = 0

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