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受欢迎的三角函数 >

sin^2(x)+sin(x)-1=0

解答

sin^{2} (x) + sin (x) - 1 = 0

解答

x = 0.66623 \dots + 2 πn, x = π - 0.66623 \dots + 2 πn

+1

度数

x = 38.17270 \dots^{\circ} + 36 0^{\circ} n, x = 141.82729 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sin^{2} (x) + sin (x) - 1 = 0

用替代法求解

sin^{2} (x) + sin (x) - 1 = 0

令

: sin (x) = u

u^{2} + u - 1 = 0

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

u^{2} + u - 1 = 0

使用求根公式求解

u^{2} + u - 1 = 0

二次方程求根公式:

若

a = 1, b = 1, c = - 1

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

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

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

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

使用法则

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot (- 1)

使用法则

- (- a) = a

= 1 + 4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 1 + 4

数字相加:

1 + 4 = 5

= 5

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

将解分隔开

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

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

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

数字相乘:

2 \cdot 1 = 2

= \frac{- 1 + 5}{2}

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

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

数字相乘:

2 \cdot 1 = 2

= \frac{- 1 - 5}{2}

二次方程组的解是:

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

u = sin (x)

代回

sin (x) = \frac{- 1 + 5}{2}, sin (x) = \frac{- 1 - 5}{2}

sin (x) = \frac{- 1 + 5}{2}, sin (x) = \frac{- 1 - 5}{2}

sin (x) = \frac{- 1 + 5}{2} : x = arcsin (\frac{- 1 + 5}{2}) + 2 πn, x = π - arcsin (\frac{- 1 + 5}{2}) + 2 πn

sin (x) = \frac{- 1 + 5}{2}

使用反三角函数性质

sin (x) = \frac{- 1 + 5}{2}

sin (x) = \frac{- 1 + 5}{2}

的通解

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

x = arcsin (\frac{- 1 + 5}{2}) + 2 πn, x = π - arcsin (\frac{- 1 + 5}{2}) + 2 πn

x = arcsin (\frac{- 1 + 5}{2}) + 2 πn, x = π - arcsin (\frac{- 1 + 5}{2}) + 2 πn

sin (x) = \frac{- 1 - 5}{2} :

无解

sin (x) = \frac{- 1 - 5}{2}

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = arcsin (\frac{- 1 + 5}{2}) + 2 πn, x = π - arcsin (\frac{- 1 + 5}{2}) + 2 πn

以小数形式表示解

x = 0.66623 \dots + 2 πn, x = π - 0.66623 \dots + 2 πn

作图

流行的例子

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

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

sin(θ)=-1/(sqrt(2))

sin (θ) = - \frac{1}{2}

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

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

tan(θ)=-2sin(θ)

tan (θ) = - 2 sin (θ)

5 cos (x) = 0