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受欢迎的三角函数 >

2cos^2(θ)-2sin^2(θ)=sqrt(3)

解答

2 cos^{2} (θ) - 2 sin^{2} (θ) = 3

解答

θ = 0.26179 \dots + 2 πn, θ = π - 0.26179 \dots + 2 πn, θ = - 0.26179 \dots + 2 πn, θ = π + 0.26179 \dots + 2 πn

+1

度数

θ = 1 5^{\circ} + 36 0^{\circ} n, θ = 16 5^{\circ} + 36 0^{\circ} n, θ = - 1 5^{\circ} + 36 0^{\circ} n, θ = 19 5^{\circ} + 36 0^{\circ} n

求解步骤

2 cos^{2} (θ) - 2 sin^{2} (θ) = 3

两边减去

3

2 cos^{2} (θ) - 2 sin^{2} (θ) - 3 = 0

使用三角恒等式改写

- 3 + 2 cos^{2} (θ) - 2 sin^{2} (θ)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

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

= - 3 + 2 (1 - sin^{2} (θ)) - 2 sin^{2} (θ)

化简

- 3 + 2 (1 - sin^{2} (θ)) - 2 sin^{2} (θ) : - 3 + 2 - 4 sin^{2} (θ)

- 3 + 2 (1 - sin^{2} (θ)) - 2 sin^{2} (θ)

乘开

2 (1 - sin^{2} (θ)) : 2 - 2 sin^{2} (θ)

2 (1 - sin^{2} (θ))

使用分配律:

a (b - c) = ab - a c

a = 2, b = 1, c = sin^{2} (θ)

= 2 \cdot 1 - 2 sin^{2} (θ)

数字相乘:

2 \cdot 1 = 2

= 2 - 2 sin^{2} (θ)

= - 3 + 2 - 2 sin^{2} (θ) - 2 sin^{2} (θ)

同类项相加:

- 2 sin^{2} (θ) - 2 sin^{2} (θ) = - 4 sin^{2} (θ)

= - 3 + 2 - 4 sin^{2} (θ)

= - 3 + 2 - 4 sin^{2} (θ)

2 - 3 - 4 sin^{2} (θ) = 0

用替代法求解

2 - 3 - 4 sin^{2} (θ) = 0

令

: sin (θ) = u

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

2 - 3 - 4 u^{2} = 0 : u = \frac{2 - 3}{2}, u = - \frac{2 - 3}{2}

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

将

2

到右边

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

两边减去

2

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

化简

- 3 - 4 u^{2} = - 2

- 3 - 4 u^{2} = - 2

将

3

到右边

- 3 - 4 u^{2} = - 2

两边加上

3

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

化简

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

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

两边除以

- 4

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

两边除以

- 4

\frac{- 4 u ^{2}}{- 4} = - \frac{2}{- 4} + \frac{3}{- 4}

化简

\frac{- 4 u ^{2}}{- 4} = - \frac{2}{- 4} + \frac{3}{- 4}

化简

\frac{- 4 u ^{2}}{- 4} : u^{2}

\frac{- 4 u ^{2}}{- 4}

使用分式法则:

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

= \frac{4 u ^{2}}{4}

数字相除:

\frac{4}{4} = 1

= u^{2}

化简

- \frac{2}{- 4} + \frac{3}{- 4} : \frac{2 - 3}{4}

- \frac{2}{- 4} + \frac{3}{- 4}

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

使用分式法则:

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

- 2 + 3 = - (2 - 3)

= \frac{2 - 3}{4}

u^{2} = \frac{2 - 3}{4}

u^{2} = \frac{2 - 3}{4}

u^{2} = \frac{2 - 3}{4}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{2 - 3}{4}, u = - \frac{2 - 3}{4}

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

\frac{2 - 3}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{2 - 3}{4}

4 = 2

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

= \frac{2 - 3}{2}

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

- \frac{2 - 3}{4}

化简

\frac{2 - 3}{4} : \frac{2 - 3}{2}

\frac{2 - 3}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{2 - 3}{4}

4 = 2

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

= \frac{2 - 3}{2}

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

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

u = sin (θ)

代回

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

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

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

sin (θ) = \frac{2 - 3}{2}

使用反三角函数性质

sin (θ) = \frac{2 - 3}{2}

sin (θ) = \frac{2 - 3}{2}

的通解

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

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

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

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

sin (θ) = - \frac{2 - 3}{2}

使用反三角函数性质

sin (θ) = - \frac{2 - 3}{2}

sin (θ) = - \frac{2 - 3}{2}

的通解

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

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

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

合并所有解

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

以小数形式表示解

θ = 0.26179 \dots + 2 πn, θ = π - 0.26179 \dots + 2 πn, θ = - 0.26179 \dots + 2 πn, θ = π + 0.26179 \dots + 2 πn

作图

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