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受欢迎的三角函数 >

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

解答

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

解答

x = \frac{3 π}{2} + 2 πn

+1

度数

x = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

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

化简

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

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

- (1 - sin^{2} (x)) : - 1 + sin^{2} (x)

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

打开括号

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

使用加减运算法则

- (- a) = a, - (a) = - a

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

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

数字相减:

3 - 1 = 2

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

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

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

用替代法求解

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

令

: sin (x) = u

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

2 + u^{2} + 3 u = 0 : u = - 1, u = - 2

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 1, b = 3, c = 2

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

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

3^{2} - 4 \cdot 1 \cdot 2 = 1

3^{2} - 4 \cdot 1 \cdot 2

数字相乘:

4 \cdot 1 \cdot 2 = 8

= 3^{2} - 8

3^{2} = 9

= 9 - 8

数字相减:

9 - 8 = 1

= 1

使用法则

1 = 1

= 1

u_{1, 2} = \frac{- 3 \pm 1}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- 3 + 1}{2 \cdot 1}, u_{2} = \frac{- 3 - 1}{2 \cdot 1}

u = \frac{- 3 + 1}{2 \cdot 1} : - 1

\frac{- 3 + 1}{2 \cdot 1}

数字相加/相减:

- 3 + 1 = - 2

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

数字相乘:

2 \cdot 1 = 2

= \frac{- 2}{2}

使用分式法则:

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

= - \frac{2}{2}

使用法则

\frac{a}{a} = 1

= - 1

u = \frac{- 3 - 1}{2 \cdot 1} : - 2

\frac{- 3 - 1}{2 \cdot 1}

数字相减:

- 3 - 1 = - 4

= \frac{- 4}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 4}{2}

使用分式法则:

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

= - \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= - 2

二次方程组的解是:

u = - 1, u = - 2

u = sin (x)

代回

sin (x) = - 1, sin (x) = - 2

sin (x) = - 1, sin (x) = - 2

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

sin (x) = - 1

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

sin (x) = - 2 :

无解

sin (x) = - 2

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = \frac{3 π}{2} + 2 πn

作图

流行的例子

tan^2(x)-3=0,0<x< pi/2

tan^{2} (x) - 3 = 0, 0 < x < \frac{π}{2}

3cot^2(x)=7csc(x)-7

3 cot^{2} (x) = 7 csc (x) - 7

sin (x) = 0.86

2sin(2θ)+sqrt(3)=0

2 sin (2 θ) + 3 = 0

sin^2(x)+4sin(x)+3=0

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