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受欢迎的三角函数 >

-6sin^2(θ)-3=sin(θ)-5

解答

- 6 sin^{2} (θ) - 3 = sin (θ) - 5

解答

θ = - 0.72972 \dots + 2 πn, θ = π + 0.72972 \dots + 2 πn, θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

+1

度数

θ = - 41.81031 \dots^{\circ} + 36 0^{\circ} n, θ = 221.81031 \dots^{\circ} + 36 0^{\circ} n, θ = 3 0^{\circ} + 36 0^{\circ} n, θ = 15 0^{\circ} + 36 0^{\circ} n

求解步骤

- 6 sin^{2} (θ) - 3 = sin (θ) - 5

用替代法求解

- 6 sin^{2} (θ) - 3 = sin (θ) - 5

令

: sin (θ) = u

- 6 u^{2} - 3 = u - 5

- 6 u^{2} - 3 = u - 5 : u = - \frac{2}{3}, u = \frac{1}{2}

- 6 u^{2} - 3 = u - 5

将

5

para o lado esquerdo

- 6 u^{2} - 3 = u - 5

两边加上

5

- 6 u^{2} - 3 + 5 = u - 5 + 5

化简

- 6 u^{2} + 2 = u

- 6 u^{2} + 2 = u

将

u

para o lado esquerdo

- 6 u^{2} + 2 = u

两边减去

u

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

化简

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 6, b = - 1, c = 2

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

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

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

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

使用法则

- (- a) = a

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

(- 1)^{2} = 1

(- 1)^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

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

= 1^{2}

使用法则

1^{a} = 1

= 1

4 \cdot 6 \cdot 2 = 48

4 \cdot 6 \cdot 2

数字相乘:

4 \cdot 6 \cdot 2 = 48

= 48

= 1 + 48

数字相加:

1 + 48 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 1 ) \pm 7}{2 ( - 6 )}

将解分隔开

u_{1} = \frac{- ( - 1 ) + 7}{2 ( - 6 )}, u_{2} = \frac{- ( - 1 ) - 7}{2 ( - 6 )}

u = \frac{- ( - 1 ) + 7}{2 ( - 6 )} : - \frac{2}{3}

\frac{- ( - 1 ) + 7}{2 ( - 6 )}

去除括号:

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

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

数字相加:

1 + 7 = 8

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

数字相乘:

2 \cdot 6 = 12

= \frac{8}{- 12}

使用分式法则:

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

= - \frac{8}{12}

约分:

4

= - \frac{2}{3}

u = \frac{- ( - 1 ) - 7}{2 ( - 6 )} : \frac{1}{2}

\frac{- ( - 1 ) - 7}{2 ( - 6 )}

去除括号:

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

= \frac{1 - 7}{- 2 \cdot 6}

数字相减:

1 - 7 = - 6

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

数字相乘:

2 \cdot 6 = 12

= \frac{- 6}{- 12}

使用分式法则:

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

= \frac{6}{12}

约分:

6

= \frac{1}{2}

二次方程组的解是:

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

u = sin (θ)

代回

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

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

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

sin (θ) = - \frac{2}{3}

使用反三角函数性质

sin (θ) = - \frac{2}{3}

sin (θ) = - \frac{2}{3}

的通解

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

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

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

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

sin (θ) = \frac{1}{2}

sin (θ) = \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}

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

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

合并所有解

θ = arcsin (- \frac{2}{3}) + 2 πn, θ = π + arcsin (\frac{2}{3}) + 2 πn, θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

以小数形式表示解

θ = - 0.72972 \dots + 2 πn, θ = π + 0.72972 \dots + 2 πn, θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

作图

流行的例子

2sqrt(3)sin(2x)-3=0

23 sin (2 x) - 3 = 0

sec (x) = \frac{3}{2}

sin (2 x) = 3

solvefor θ,sin(2θ)= 1/2

so l v e f or θ, sin (2 θ) = \frac{1}{2}

sin(x)+cos(x)=(sqrt(6))/2

sin (x) + cos (x) = \frac{6}{2}