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受欢迎的三角函数 >

sin^2(x)+5sin(x)=3

解答

sin^{2} (x) + 5 sin (x) = 3

解答

x = 0.57207 \dots + 2 πn, x = π - 0.57207 \dots + 2 πn

+1

度数

x = 32.77771 \dots^{\circ} + 36 0^{\circ} n, x = 147.22228 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sin^{2} (x) + 5 sin (x) = 3

用替代法求解

sin^{2} (x) + 5 sin (x) = 3

令

: sin (x) = u

u^{2} + 5 u = 3

u^{2} + 5 u = 3 : u = \frac{- 5 + 37}{2}, u = \frac{- 5 - 37}{2}

u^{2} + 5 u = 3

将

3

para o lado esquerdo

u^{2} + 5 u = 3

两边减去

3

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

化简

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 1, b = 5, c = - 3

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

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

5^{2} - 4 \cdot 1 \cdot (- 3) = 37

5^{2} - 4 \cdot 1 \cdot (- 3)

使用法则

- (- a) = a

= 5^{2} + 4 \cdot 1 \cdot 3

数字相乘:

4 \cdot 1 \cdot 3 = 12

= 5^{2} + 12

5^{2} = 25

= 25 + 12

数字相加:

25 + 12 = 37

= 37

u_{1, 2} = \frac{- 5 \pm 37}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- 5 + 37}{2 \cdot 1}, u_{2} = \frac{- 5 - 37}{2 \cdot 1}

u = \frac{- 5 + 37}{2 \cdot 1} : \frac{- 5 + 37}{2}

\frac{- 5 + 37}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 5 + 37}{2}

u = \frac{- 5 - 37}{2 \cdot 1} : \frac{- 5 - 37}{2}

\frac{- 5 - 37}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 5 - 37}{2}

二次方程组的解是:

u = \frac{- 5 + 37}{2}, u = \frac{- 5 - 37}{2}

u = sin (x)

代回

sin (x) = \frac{- 5 + 37}{2}, sin (x) = \frac{- 5 - 37}{2}

sin (x) = \frac{- 5 + 37}{2}, sin (x) = \frac{- 5 - 37}{2}

sin (x) = \frac{- 5 + 37}{2} : x = arcsin (\frac{- 5 + 37}{2}) + 2 πn, x = π - arcsin (\frac{- 5 + 37}{2}) + 2 πn

sin (x) = \frac{- 5 + 37}{2}

使用反三角函数性质

sin (x) = \frac{- 5 + 37}{2}

sin (x) = \frac{- 5 + 37}{2}

的通解

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

x = arcsin (\frac{- 5 + 37}{2}) + 2 πn, x = π - arcsin (\frac{- 5 + 37}{2}) + 2 πn

x = arcsin (\frac{- 5 + 37}{2}) + 2 πn, x = π - arcsin (\frac{- 5 + 37}{2}) + 2 πn

sin (x) = \frac{- 5 - 37}{2} :

无解

sin (x) = \frac{- 5 - 37}{2}

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = arcsin (\frac{- 5 + 37}{2}) + 2 πn, x = π - arcsin (\frac{- 5 + 37}{2}) + 2 πn

以小数形式表示解

x = 0.57207 \dots + 2 πn, x = π - 0.57207 \dots + 2 πn

作图

流行的例子

14+7cos(2x)=21cos(x)

14 + 7 cos (2 x) = 21 cos (x)

sin(3x)=0,0<= x<2pi

sin (3 x) = 0, 0 \leq x < 2 π

sin(θ)=-1/5 ,pi<θ<(3pi)/2 ,cos(θ)

sin (θ) = - \frac{1}{5}, π < θ < \frac{3 π}{2}, cos (θ)

(sin(x))/(37)=(sin(30))/(30)

\frac{sin ( x )}{37} = \frac{sin ( 3 0 ^{\circ} )}{30}

solvefor t,x=arccos(1/(sqrt(1+t^2)))

so l v e f or t, x = arccos (\frac{1}{1 + t ^{2}})