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受欢迎的三角函数 >

8sin(x)=4cos^2(x)-7

解答

8 sin (x) = 4 cos^{2} (x) - 7

解答

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

度数

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

8 sin (x) = 4 cos^{2} (x) - 7

两边减去

4 cos^{2} (x) - 7

8 sin (x) - 4 cos^{2} (x) + 7 = 0

使用三角恒等式改写

7 - 4 cos^{2} (x) + 8 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)

= 7 - 4 (1 - sin^{2} (x)) + 8 sin (x)

化简

7 - 4 (1 - sin^{2} (x)) + 8 sin (x) : 4 sin^{2} (x) + 8 sin (x) + 3

7 - 4 (1 - sin^{2} (x)) + 8 sin (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)

= 7 - 4 + 4 sin^{2} (x) + 8 sin (x)

数字相减:

7 - 4 = 3

= 4 sin^{2} (x) + 8 sin (x) + 3

= 4 sin^{2} (x) + 8 sin (x) + 3

3 + 4 sin^{2} (x) + 8 sin (x) = 0

用替代法求解

3 + 4 sin^{2} (x) + 8 sin (x) = 0

令

: sin (x) = u

3 + 4 u^{2} + 8 u = 0

3 + 4 u^{2} + 8 u = 0 : u = - \frac{1}{2}, u = - \frac{3}{2}

3 + 4 u^{2} + 8 u = 0

改写成标准形式

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

4 u^{2} + 8 u + 3 = 0

使用求根公式求解

4 u^{2} + 8 u + 3 = 0

二次方程求根公式:

若

a = 4, b = 8, c = 3

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 4 \cdot 3}{2 \cdot 4}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 4 \cdot 3}{2 \cdot 4}

8^{2} - 4 \cdot 4 \cdot 3 = 4

8^{2} - 4 \cdot 4 \cdot 3

数字相乘:

4 \cdot 4 \cdot 3 = 48

= 8^{2} - 48

8^{2} = 64

= 64 - 48

数字相减:

64 - 48 = 16

= 16

因式分解数字:

16 = 4^{2}

= 4^{2}

使用根式运算法则:

4^{2} = 4

= 4

u_{1, 2} = \frac{- 8 \pm 4}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- 8 + 4}{2 \cdot 4}, u_{2} = \frac{- 8 - 4}{2 \cdot 4}

u = \frac{- 8 + 4}{2 \cdot 4} : - \frac{1}{2}

\frac{- 8 + 4}{2 \cdot 4}

数字相加/相减:

- 8 + 4 = - 4

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

数字相乘:

2 \cdot 4 = 8

= \frac{- 4}{8}

使用分式法则:

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

= - \frac{4}{8}

约分:

4

= - \frac{1}{2}

u = \frac{- 8 - 4}{2 \cdot 4} : - \frac{3}{2}

\frac{- 8 - 4}{2 \cdot 4}

数字相减:

- 8 - 4 = - 12

= \frac{- 12}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 12}{8}

使用分式法则:

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

= - \frac{12}{8}

约分:

4

= - \frac{3}{2}

二次方程组的解是:

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

u = sin (x)

代回

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

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

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

无解

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

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

作图

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