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10sin(x)=4cos^2(x)+2

解答

10 sin (x) = 4 cos^{2} (x) + 2

解答

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

求解步骤

10 sin (x) = 4 cos^{2} (x) + 2

两边减去

4 cos^{2} (x) + 2

10 sin (x) - 4 cos^{2} (x) - 2 = 0

使用三角恒等式改写

- 2 + 10 sin (x) - 4 cos^{2} (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)

= - 2 + 10 sin (x) - 4 (1 - sin^{2} (x))

化简

- 2 + 10 sin (x) - 4 (1 - sin^{2} (x)) : 4 sin^{2} (x) + 10 sin (x) - 6

- 2 + 10 sin (x) - 4 (1 - sin^{2} (x))

乘开

- 4 (1 - sin^{2} (x)) : - 4 + 4 sin^{2} (x)

- 4 (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = - 4, b = 1, c = sin^{2} (x)

= - 4 \cdot 1 - (- 4) sin^{2} (x)

使用加减运算法则

- (- a) = a

= - 4 \cdot 1 + 4 sin^{2} (x)

数字相乘:

4 \cdot 1 = 4

= - 4 + 4 sin^{2} (x)

= - 2 + 10 sin (x) - 4 + 4 sin^{2} (x)

化简

- 2 + 10 sin (x) - 4 + 4 sin^{2} (x) : 4 sin^{2} (x) + 10 sin (x) - 6

- 2 + 10 sin (x) - 4 + 4 sin^{2} (x)

对同类项分组

= 10 sin (x) + 4 sin^{2} (x) - 2 - 4

数字相减:

- 2 - 4 = - 6

= 4 sin^{2} (x) + 10 sin (x) - 6

= 4 sin^{2} (x) + 10 sin (x) - 6

= 4 sin^{2} (x) + 10 sin (x) - 6

- 6 + 10 sin (x) + 4 sin^{2} (x) = 0

用替代法求解

- 6 + 10 sin (x) + 4 sin^{2} (x) = 0

令

: sin (x) = u

- 6 + 10 u + 4 u^{2} = 0

- 6 + 10 u + 4 u^{2} = 0 : u = \frac{1}{2}, u = - 3

- 6 + 10 u + 4 u^{2} = 0

改写成标准形式

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

4 u^{2} + 10 u - 6 = 0

使用求根公式求解

4 u^{2} + 10 u - 6 = 0

二次方程求根公式:

若

a = 4, b = 10, c = - 6

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 \cdot 4 ( - 6 )}{2 \cdot 4}

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 \cdot 4 ( - 6 )}{2 \cdot 4}

1 0^{2} - 4 \cdot 4 (- 6) = 14

1 0^{2} - 4 \cdot 4 (- 6)

使用法则

- (- a) = a

= 1 0^{2} + 4 \cdot 4 \cdot 6

数字相乘:

4 \cdot 4 \cdot 6 = 96

= 1 0^{2} + 96

1 0^{2} = 100

= 100 + 96

数字相加:

100 + 96 = 196

= 196

因式分解数字:

196 = 1 4^{2}

= 1 4^{2}

使用根式运算法则:

1 4^{2} = 14

= 14

u_{1, 2} = \frac{- 10 \pm 14}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- 10 + 14}{2 \cdot 4}, u_{2} = \frac{- 10 - 14}{2 \cdot 4}

u = \frac{- 10 + 14}{2 \cdot 4} : \frac{1}{2}

\frac{- 10 + 14}{2 \cdot 4}

数字相加/相减:

- 10 + 14 = 4

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

数字相乘:

2 \cdot 4 = 8

= \frac{4}{8}

约分:

4

= \frac{1}{2}

u = \frac{- 10 - 14}{2 \cdot 4} : - 3

\frac{- 10 - 14}{2 \cdot 4}

数字相减:

- 10 - 14 = - 24

= \frac{- 24}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 24}{8}

使用分式法则:

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

= - \frac{24}{8}

数字相除:

\frac{24}{8} = 3

= - 3

二次方程组的解是:

u = \frac{1}{2}, u = - 3

u = sin (x)

代回

sin (x) = \frac{1}{2}, sin (x) = - 3

sin (x) = \frac{1}{2}, sin (x) = - 3

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

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{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = - 3 :

无解

sin (x) = - 3

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

流行的例子

-4+sin(-3x+(2pi)/3)=-4

- 4 + sin (- 3 x + \frac{2 π}{3}) = - 4

2*sin(x)*cos(x)=cos(x)

2 \cdot sin (x) \cdot cos (x) = cos (x)

sin(2θ)=(-2sqrt(6))/7

sin (2 θ) = \frac{- 2 6}{7}

solvefor x,sin(x)=-7/25 求解

x, sin (x) = - \frac{7}{25}

1/(tan(α))+tan(α)= 1/(sin(α))

\frac{1}{tan ( α )} + tan (α) = \frac{1}{sin ( α )}