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受欢迎的三角函数 >

3sin^2(θ)-8sin(θ)+5=0

解答

3 sin^{2} (θ) - 8 sin (θ) + 5 = 0

解答

θ = \frac{π}{2} + 2 πn

+1

度数

θ = 9 0^{\circ} + 36 0^{\circ} n

求解步骤

3 sin^{2} (θ) - 8 sin (θ) + 5 = 0

用替代法求解

3 sin^{2} (θ) - 8 sin (θ) + 5 = 0

令

: sin (θ) = u

3 u^{2} - 8 u + 5 = 0

3 u^{2} - 8 u + 5 = 0 : u = \frac{5}{3}, u = 1

3 u^{2} - 8 u + 5 = 0

使用求根公式求解

3 u^{2} - 8 u + 5 = 0

二次方程求根公式:

若

a = 3, b = - 8, c = 5

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 3 \cdot 5}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 3 \cdot 5}{2 \cdot 3}

(- 8)^{2} - 4 \cdot 3 \cdot 5 = 2

(- 8)^{2} - 4 \cdot 3 \cdot 5

使用指数法则:

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

若

n

是偶数

(- 8)^{2} = 8^{2}

= 8^{2} - 4 \cdot 3 \cdot 5

数字相乘:

4 \cdot 3 \cdot 5 = 60

= 8^{2} - 60

8^{2} = 64

= 64 - 60

数字相减:

64 - 60 = 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{- ( - 8 ) \pm 2}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 8 ) + 2}{2 \cdot 3}, u_{2} = \frac{- ( - 8 ) - 2}{2 \cdot 3}

u = \frac{- ( - 8 ) + 2}{2 \cdot 3} : \frac{5}{3}

\frac{- ( - 8 ) + 2}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{8 + 2}{2 \cdot 3}

数字相加:

8 + 2 = 10

= \frac{10}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{10}{6}

约分:

2

= \frac{5}{3}

u = \frac{- ( - 8 ) - 2}{2 \cdot 3} : 1

\frac{- ( - 8 ) - 2}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{8 - 2}{2 \cdot 3}

数字相减:

8 - 2 = 6

= \frac{6}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{6}{6}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

u = \frac{5}{3}, u = 1

u = sin (θ)

代回

sin (θ) = \frac{5}{3}, sin (θ) = 1

sin (θ) = \frac{5}{3}, sin (θ) = 1

sin (θ) = \frac{5}{3} :

无解

sin (θ) = \frac{5}{3}

- 1 \leq sin (x) \leq 1

无解

sin (θ) = 1 : θ = \frac{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{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

合并所有解

θ = \frac{π}{2} + 2 πn

作图

流行的例子

8 cos (x) - 4 = 0

sin^2(θ)=5sin(θ)+6

sin^{2} (θ) = 5 sin (θ) + 6

2 = 2 sec (x)

cos(x)-sin(x)=(sqrt(2))/2

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

2/5 cos(x)=(sqrt(2))/5

\frac{2}{5} cos (x) = \frac{2}{5}