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受欢迎的三角函数 >

6sin^2(θ)+5sin(θ)=-2sin(θ)-1

解答

6 sin^{2} (θ) + 5 sin (θ) = - 2 sin (θ) - 1

解答

θ = - 0.16744 \dots + 2 πn, θ = π + 0.16744 \dots + 2 πn, θ = \frac{3 π}{2} + 2 πn

+1

度数

θ = - 9.59406 \dots^{\circ} + 36 0^{\circ} n, θ = 189.59406 \dots^{\circ} + 36 0^{\circ} n, θ = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

6 sin^{2} (θ) + 5 sin (θ) = - 2 sin (θ) - 1

用替代法求解

6 sin^{2} (θ) + 5 sin (θ) = - 2 sin (θ) - 1

令

: sin (θ) = u

6 u^{2} + 5 u = - 2 u - 1

6 u^{2} + 5 u = - 2 u - 1 : u = - \frac{1}{6}, u = - 1

6 u^{2} + 5 u = - 2 u - 1

将

1

para o lado esquerdo

6 u^{2} + 5 u = - 2 u - 1

两边加上

1

6 u^{2} + 5 u + 1 = - 2 u - 1 + 1

化简

6 u^{2} + 5 u + 1 = - 2 u

6 u^{2} + 5 u + 1 = - 2 u

将

2 u

para o lado esquerdo

6 u^{2} + 5 u + 1 = - 2 u

两边加上

2 u

6 u^{2} + 5 u + 1 + 2 u = - 2 u + 2 u

化简

6 u^{2} + 7 u + 1 = 0

6 u^{2} + 7 u + 1 = 0

使用求根公式求解

6 u^{2} + 7 u + 1 = 0

二次方程求根公式:

若

a = 6, b = 7, c = 1

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 6 \cdot 1}{2 \cdot 6}

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 6 \cdot 1}{2 \cdot 6}

7^{2} - 4 \cdot 6 \cdot 1 = 5

7^{2} - 4 \cdot 6 \cdot 1

数字相乘:

4 \cdot 6 \cdot 1 = 24

= 7^{2} - 24

7^{2} = 49

= 49 - 24

数字相减:

49 - 24 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- 7 \pm 5}{2 \cdot 6}

将解分隔开

u_{1} = \frac{- 7 + 5}{2 \cdot 6}, u_{2} = \frac{- 7 - 5}{2 \cdot 6}

u = \frac{- 7 + 5}{2 \cdot 6} : - \frac{1}{6}

\frac{- 7 + 5}{2 \cdot 6}

数字相加/相减:

- 7 + 5 = - 2

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

数字相乘:

2 \cdot 6 = 12

= \frac{- 2}{12}

使用分式法则:

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

= - \frac{2}{12}

约分:

2

= - \frac{1}{6}

u = \frac{- 7 - 5}{2 \cdot 6} : - 1

\frac{- 7 - 5}{2 \cdot 6}

数字相减:

- 7 - 5 = - 12

= \frac{- 12}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{- 12}{12}

使用分式法则:

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

= - \frac{12}{12}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

u = - \frac{1}{6}, u = - 1

u = sin (θ)

代回

sin (θ) = - \frac{1}{6}, sin (θ) = - 1

sin (θ) = - \frac{1}{6}, sin (θ) = - 1

sin (θ) = - \frac{1}{6} : θ = arcsin (- \frac{1}{6}) + 2 πn, θ = π + arcsin (\frac{1}{6}) + 2 πn

sin (θ) = - \frac{1}{6}

使用反三角函数性质

sin (θ) = - \frac{1}{6}

sin (θ) = - \frac{1}{6}

的通解

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

θ = arcsin (- \frac{1}{6}) + 2 πn, θ = π + arcsin (\frac{1}{6}) + 2 πn

θ = arcsin (- \frac{1}{6}) + 2 πn, θ = π + arcsin (\frac{1}{6}) + 2 πn

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

sin (θ) = - 1

sin (θ) = - 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}

θ = \frac{3 π}{2} + 2 πn

θ = \frac{3 π}{2} + 2 πn

合并所有解

θ = arcsin (- \frac{1}{6}) + 2 πn, θ = π + arcsin (\frac{1}{6}) + 2 πn, θ = \frac{3 π}{2} + 2 πn

以小数形式表示解

θ = - 0.16744 \dots + 2 πn, θ = π + 0.16744 \dots + 2 πn, θ = \frac{3 π}{2} + 2 πn

作图

流行的例子

solvefor t,x=5cos(t)

so l v e f or t, x = 5 cos (t)

tan (θ) = \frac{π}{2}

49sin(2β)=25,tan(β)<0

49 sin (2 β) = 25, tan (β) < 0

cos (θ) = - 0.076

4cos^2(θ)=2,0<= θ<= 2pi

4 cos^{2} (θ) = 2, 0 \leq θ \leq 2 π