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受欢迎的三角函数 >

2sin^2(x)=3-5cos(x)

解答

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

解答

x = 1.34977 \dots + 2 πn, x = 2 π - 1.34977 \dots + 2 πn

+1

度数

x = 77.33656 \dots^{\circ} + 36 0^{\circ} n, x = 282.66343 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

3 - 5 cos (x)

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

化简

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

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

乘开

2 (1 - cos^{2} (x)) : 2 - 2 cos^{2} (x)

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

使用分配律:

a (b - c) = ab - a c

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

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

数字相乘:

2 \cdot 1 = 2

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

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

数字相加/相减:

- 3 + 2 = - 1

= 5 cos (x) - 2 cos^{2} (x) - 1

= 5 cos (x) - 2 cos^{2} (x) - 1

- 1 - 2 cos^{2} (x) + 5 cos (x) = 0

用替代法求解

- 1 - 2 cos^{2} (x) + 5 cos (x) = 0

令

: cos (x) = u

- 1 - 2 u^{2} + 5 u = 0

- 1 - 2 u^{2} + 5 u = 0 : u = \frac{5 - 17}{4}, u = \frac{5 + 17}{4}

- 1 - 2 u^{2} + 5 u = 0

改写成标准形式

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

- 2 u^{2} + 5 u - 1 = 0

使用求根公式求解

- 2 u^{2} + 5 u - 1 = 0

二次方程求根公式:

若

a = - 2, b = 5, c = - 1

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

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

5^{2} - 4 (- 2) (- 1) = 17

5^{2} - 4 (- 2) (- 1)

使用法则

- (- a) = a

= 5^{2} - 4 \cdot 2 \cdot 1

数字相乘:

4 \cdot 2 \cdot 1 = 8

= 5^{2} - 8

5^{2} = 25

= 25 - 8

数字相减:

25 - 8 = 17

= 17

u_{1, 2} = \frac{- 5 \pm 17}{2 ( - 2 )}

将解分隔开

u_{1} = \frac{- 5 + 17}{2 ( - 2 )}, u_{2} = \frac{- 5 - 17}{2 ( - 2 )}

u = \frac{- 5 + 17}{2 ( - 2 )} : \frac{5 - 17}{4}

\frac{- 5 + 17}{2 ( - 2 )}

去除括号:

(- a) = - a

= \frac{- 5 + 17}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 5 + 17}{- 4}

使用分式法则:

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

- 5 + 17 = - (5 - 17)

= \frac{5 - 17}{4}

u = \frac{- 5 - 17}{2 ( - 2 )} : \frac{5 + 17}{4}

\frac{- 5 - 17}{2 ( - 2 )}

去除括号:

(- a) = - a

= \frac{- 5 - 17}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 5 - 17}{- 4}

使用分式法则:

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

- 5 - 17 = - (5 + 17)

= \frac{5 + 17}{4}

二次方程组的解是:

u = \frac{5 - 17}{4}, u = \frac{5 + 17}{4}

u = cos (x)

代回

cos (x) = \frac{5 - 17}{4}, cos (x) = \frac{5 + 17}{4}

cos (x) = \frac{5 - 17}{4}, cos (x) = \frac{5 + 17}{4}

cos (x) = \frac{5 - 17}{4} : x = arccos (\frac{5 - 17}{4}) + 2 πn, x = 2 π - arccos (\frac{5 - 17}{4}) + 2 πn

cos (x) = \frac{5 - 17}{4}

使用反三角函数性质

cos (x) = \frac{5 - 17}{4}

cos (x) = \frac{5 - 17}{4}

的通解

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

x = arccos (\frac{5 - 17}{4}) + 2 πn, x = 2 π - arccos (\frac{5 - 17}{4}) + 2 πn

x = arccos (\frac{5 - 17}{4}) + 2 πn, x = 2 π - arccos (\frac{5 - 17}{4}) + 2 πn

cos (x) = \frac{5 + 17}{4} :

无解

cos (x) = \frac{5 + 17}{4}

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = arccos (\frac{5 - 17}{4}) + 2 πn, x = 2 π - arccos (\frac{5 - 17}{4}) + 2 πn

以小数形式表示解

x = 1.34977 \dots + 2 πn, x = 2 π - 1.34977 \dots + 2 πn

作图

流行的例子

-3cos(2θ)-18sin(θ)-8=-5sin(θ)-6

- 3 cos (2 θ) - 18 sin (θ) - 8 = - 5 sin (θ) - 6

tan^2(x)+4tan(x)=3

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

sin (1 - 2 x) = 1

5cos(x)+2=cos(x)

5 cos (x) + 2 = cos (x)

sec(x)=-5,pi<x<(3pi)/2

sec (x) = - 5, π < x < \frac{3 π}{2}