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受欢迎的三角函数 >

8sin^2(x)+6sin(x)+1=0

解答

8 sin^{2} (x) + 6 sin (x) + 1 = 0

解答

x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

度数

x = - 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 194.47751 \dots^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

8 sin^{2} (x) + 6 sin (x) + 1 = 0

用替代法求解

8 sin^{2} (x) + 6 sin (x) + 1 = 0

令

: sin (x) = u

8 u^{2} + 6 u + 1 = 0

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

8 u^{2} + 6 u + 1 = 0

使用求根公式求解

8 u^{2} + 6 u + 1 = 0

二次方程求根公式:

若

a = 8, b = 6, c = 1

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 8 \cdot 1}{2 \cdot 8}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 8 \cdot 1}{2 \cdot 8}

6^{2} - 4 \cdot 8 \cdot 1 = 2

6^{2} - 4 \cdot 8 \cdot 1

数字相乘:

4 \cdot 8 \cdot 1 = 32

= 6^{2} - 32

6^{2} = 36

= 36 - 32

数字相减:

36 - 32 = 4

= 4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

u_{1, 2} = \frac{- 6 \pm 2}{2 \cdot 8}

将解分隔开

u_{1} = \frac{- 6 + 2}{2 \cdot 8}, u_{2} = \frac{- 6 - 2}{2 \cdot 8}

u = \frac{- 6 + 2}{2 \cdot 8} : - \frac{1}{4}

\frac{- 6 + 2}{2 \cdot 8}

数字相加/相减:

- 6 + 2 = - 4

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

数字相乘:

2 \cdot 8 = 16

= \frac{- 4}{16}

使用分式法则:

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

= - \frac{4}{16}

约分:

4

= - \frac{1}{4}

u = \frac{- 6 - 2}{2 \cdot 8} : - \frac{1}{2}

\frac{- 6 - 2}{2 \cdot 8}

数字相减:

- 6 - 2 = - 8

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

数字相乘:

2 \cdot 8 = 16

= \frac{- 8}{16}

使用分式法则:

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

= - \frac{8}{16}

约分:

8

= - \frac{1}{2}

二次方程组的解是:

u = - \frac{1}{4}, u = - \frac{1}{2}

u = sin (x)

代回

sin (x) = - \frac{1}{4}, sin (x) = - \frac{1}{2}

sin (x) = - \frac{1}{4}, sin (x) = - \frac{1}{2}

sin (x) = - \frac{1}{4} : x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

sin (x) = - \frac{1}{4}

使用反三角函数性质

sin (x) = - \frac{1}{4}

sin (x) = - \frac{1}{4}

的通解

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

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

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

合并所有解

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

以小数形式表示解

x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

作图

流行的例子

sin (\frac{θ}{2}) = 0

tan(θ/2-pi/4)=1

tan (\frac{θ}{2} - \frac{π}{4}) = 1

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

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3 tan (x) = 3

3cos^2(x)-8cos(x)-3=0

3 cos^{2} (x) - 8 cos (x) - 3 = 0