zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

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

解答

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

解答

x = 0.45387 \dots + 2 πn, x = π - 0.45387 \dots + 2 πn

+1

度数

x = 26.00484 \dots^{\circ} + 36 0^{\circ} n, x = 153.99515 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

3 - 5 sin (x)

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

使用三角恒等式改写

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

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

化简

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

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

用替代法求解

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

令

: sin (x) = u

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

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

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

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

使用法则

- (- a) = a

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

数字相乘:

4 \cdot 1 \cdot 2 = 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 ( - 1 )}

将解分隔开

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

u = \frac{- 5 + 17}{2 ( - 1 )} : \frac{5 - 17}{2}

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

去除括号:

(- a) = - a

= \frac{- 5 + 17}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

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

使用分式法则:

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

- 5 + 17 = - (5 - 17)

= \frac{5 - 17}{2}

u = \frac{- 5 - 17}{2 ( - 1 )} : \frac{5 + 17}{2}

\frac{- 5 - 17}{2 ( - 1 )}

去除括号:

(- a) = - a

= \frac{- 5 - 17}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

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

使用分式法则:

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

- 5 - 17 = - (5 + 17)

= \frac{5 + 17}{2}

二次方程组的解是:

u = \frac{5 - 17}{2}, u = \frac{5 + 17}{2}

u = sin (x)

代回

sin (x) = \frac{5 - 17}{2}, sin (x) = \frac{5 + 17}{2}

sin (x) = \frac{5 - 17}{2}, sin (x) = \frac{5 + 17}{2}

sin (x) = \frac{5 - 17}{2} : x = arcsin (\frac{5 - 17}{2}) + 2 πn, x = π - arcsin (\frac{5 - 17}{2}) + 2 πn

sin (x) = \frac{5 - 17}{2}

使用反三角函数性质

sin (x) = \frac{5 - 17}{2}

sin (x) = \frac{5 - 17}{2}

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

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

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

sin (x) = \frac{5 + 17}{2} :

无解

sin (x) = \frac{5 + 17}{2}

- 1 \leq sin (x) \leq 1

无解

合并所有解

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

以小数形式表示解

x = 0.45387 \dots + 2 πn, x = π - 0.45387 \dots + 2 πn

作图

流行的例子

8sin(x/2)+8cos(x)=0

8 sin (\frac{x}{2}) + 8 cos (x) = 0

tan(θ/2+pi/6)=1

tan (\frac{θ}{2} + \frac{π}{6}) = 1

sec(x)=-sqrt(2),0<x<2pi

sec (x) = - 2, 0 < x < 2 π

cos (x) = \frac{7}{16}

4 sin (3 x) = 2