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受欢迎的三角函数 >

5cos^2(x)+6cos(x)=8

解答

5 cos^{2} (x) + 6 cos (x) = 8

解答

x = 0.64350 \dots + 2 πn, x = 2 π - 0.64350 \dots + 2 πn

+1

度数

x = 36.86989 \dots^{\circ} + 36 0^{\circ} n, x = 323.13010 \dots^{\circ} + 36 0^{\circ} n

求解步骤

5 cos^{2} (x) + 6 cos (x) = 8

用替代法求解

5 cos^{2} (x) + 6 cos (x) = 8

令

: cos (x) = u

5 u^{2} + 6 u = 8

5 u^{2} + 6 u = 8 : u = \frac{4}{5}, u = - 2

5 u^{2} + 6 u = 8

将

8

para o lado esquerdo

5 u^{2} + 6 u = 8

两边减去

8

5 u^{2} + 6 u - 8 = 8 - 8

化简

5 u^{2} + 6 u - 8 = 0

5 u^{2} + 6 u - 8 = 0

使用求根公式求解

5 u^{2} + 6 u - 8 = 0

二次方程求根公式:

若

a = 5, b = 6, c = - 8

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 5 ( - 8 )}{2 \cdot 5}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 5 ( - 8 )}{2 \cdot 5}

6^{2} - 4 \cdot 5 (- 8) = 14

6^{2} - 4 \cdot 5 (- 8)

使用法则

- (- a) = a

= 6^{2} + 4 \cdot 5 \cdot 8

数字相乘:

4 \cdot 5 \cdot 8 = 160

= 6^{2} + 160

6^{2} = 36

= 36 + 160

数字相加:

36 + 160 = 196

= 196

因式分解数字:

196 = 1 4^{2}

= 1 4^{2}

使用根式运算法则:

n a^{n} = a

1 4^{2} = 14

= 14

u_{1, 2} = \frac{- 6 \pm 14}{2 \cdot 5}

将解分隔开

u_{1} = \frac{- 6 + 14}{2 \cdot 5}, u_{2} = \frac{- 6 - 14}{2 \cdot 5}

u = \frac{- 6 + 14}{2 \cdot 5} : \frac{4}{5}

\frac{- 6 + 14}{2 \cdot 5}

数字相加/相减:

- 6 + 14 = 8

= \frac{8}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{8}{10}

约分:

2

= \frac{4}{5}

u = \frac{- 6 - 14}{2 \cdot 5} : - 2

\frac{- 6 - 14}{2 \cdot 5}

数字相减:

- 6 - 14 = - 20

= \frac{- 20}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{- 20}{10}

使用分式法则:

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

= - \frac{20}{10}

数字相除:

\frac{20}{10} = 2

= - 2

二次方程组的解是:

u = \frac{4}{5}, u = - 2

u = cos (x)

代回

cos (x) = \frac{4}{5}, cos (x) = - 2

cos (x) = \frac{4}{5}, cos (x) = - 2

cos (x) = \frac{4}{5} : x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn

cos (x) = \frac{4}{5}

使用反三角函数性质

cos (x) = \frac{4}{5}

cos (x) = \frac{4}{5}

的通解

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

x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn

x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn

cos (x) = - 2 :

无解

cos (x) = - 2

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn

以小数形式表示解

x = 0.64350 \dots + 2 πn, x = 2 π - 0.64350 \dots + 2 πn

作图

流行的例子

tan (x) = \frac{- 4}{3}

sin (x) = \frac{2.5}{5}

3 cos^{2} (x) = 0

tan(4x)=2tan(2x)

tan (4 x) = 2 tan (2 x)

sec(x)-4=-sec(x)

sec (x) - 4 = - sec (x)