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受欢迎的三角函数 >

-4sin^2(θ)-3sin(θ)+5=0

解答

- 4 sin^{2} (θ) - 3 sin (θ) + 5 = 0

解答

θ = 0.93440 \dots + 2 πn, θ = π - 0.93440 \dots + 2 πn

+1

度数

θ = 53.53765 \dots^{\circ} + 36 0^{\circ} n, θ = 126.46234 \dots^{\circ} + 36 0^{\circ} n

求解步骤

- 4 sin^{2} (θ) - 3 sin (θ) + 5 = 0

用替代法求解

- 4 sin^{2} (θ) - 3 sin (θ) + 5 = 0

令

: sin (θ) = u

- 4 u^{2} - 3 u + 5 = 0

- 4 u^{2} - 3 u + 5 = 0 : u = - \frac{3 + 89}{8}, u = \frac{89 - 3}{8}

- 4 u^{2} - 3 u + 5 = 0

使用求根公式求解

- 4 u^{2} - 3 u + 5 = 0

二次方程求根公式:

若

a = - 4, b = - 3, c = 5

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 4 ) \cdot 5}{2 ( - 4 )}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 ( - 4 ) \cdot 5}{2 ( - 4 )}

(- 3)^{2} - 4 (- 4) \cdot 5 = 89

(- 3)^{2} - 4 (- 4) \cdot 5

使用法则

- (- a) = a

= (- 3)^{2} + 4 \cdot 4 \cdot 5

使用指数法则:

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

若

n

是偶数

(- 3)^{2} = 3^{2}

= 3^{2} + 4 \cdot 4 \cdot 5

数字相乘:

4 \cdot 4 \cdot 5 = 80

= 3^{2} + 80

3^{2} = 9

= 9 + 80

数字相加:

9 + 80 = 89

= 89

u_{1, 2} = \frac{- ( - 3 ) \pm 89}{2 ( - 4 )}

将解分隔开

u_{1} = \frac{- ( - 3 ) + 89}{2 ( - 4 )}, u_{2} = \frac{- ( - 3 ) - 89}{2 ( - 4 )}

u = \frac{- ( - 3 ) + 89}{2 ( - 4 )} : - \frac{3 + 89}{8}

\frac{- ( - 3 ) + 89}{2 ( - 4 )}

去除括号:

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

= \frac{3 + 89}{- 2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{3 + 89}{- 8}

使用分式法则:

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

= - \frac{3 + 89}{8}

u = \frac{- ( - 3 ) - 89}{2 ( - 4 )} : \frac{89 - 3}{8}

\frac{- ( - 3 ) - 89}{2 ( - 4 )}

去除括号:

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

= \frac{3 - 89}{- 2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{3 - 89}{- 8}

使用分式法则:

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

3 - 89 = - (89 - 3)

= \frac{89 - 3}{8}

二次方程组的解是:

u = - \frac{3 + 89}{8}, u = \frac{89 - 3}{8}

u = sin (θ)

代回

sin (θ) = - \frac{3 + 89}{8}, sin (θ) = \frac{89 - 3}{8}

sin (θ) = - \frac{3 + 89}{8}, sin (θ) = \frac{89 - 3}{8}

sin (θ) = - \frac{3 + 89}{8} :

无解

sin (θ) = - \frac{3 + 89}{8}

- 1 \leq sin (x) \leq 1

无解

sin (θ) = \frac{89 - 3}{8} : θ = arcsin (\frac{89 - 3}{8}) + 2 πn, θ = π - arcsin (\frac{89 - 3}{8}) + 2 πn

sin (θ) = \frac{89 - 3}{8}

使用反三角函数性质

sin (θ) = \frac{89 - 3}{8}

sin (θ) = \frac{89 - 3}{8}

的通解

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

θ = arcsin (\frac{89 - 3}{8}) + 2 πn, θ = π - arcsin (\frac{89 - 3}{8}) + 2 πn

θ = arcsin (\frac{89 - 3}{8}) + 2 πn, θ = π - arcsin (\frac{89 - 3}{8}) + 2 πn

合并所有解

θ = arcsin (\frac{89 - 3}{8}) + 2 πn, θ = π - arcsin (\frac{89 - 3}{8}) + 2 πn

以小数形式表示解

θ = 0.93440 \dots + 2 πn, θ = π - 0.93440 \dots + 2 πn

作图

流行的例子

cos^2(x)-sin^2(x)=-1/2

cos^{2} (x) - sin^{2} (x) = - \frac{1}{2}

cos(θ)+2cos(θ+240)=1

cos (θ) + 2 cos (θ + 24 0^{\circ}) = 1

arccos(x)= pi/8

arccos (x) = \frac{π}{8}

sin (θ) = 5

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

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