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受欢迎的三角函数 >

9cos(2θ)=9cos^2(θ)-4

解答

9 cos (2 θ) = 9 cos^{2} (θ) - 4

解答

θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn, θ = - 0.72972 \dots + 2 πn, θ = π + 0.72972 \dots + 2 πn

+1

度数

θ = 41.81031 \dots^{\circ} + 36 0^{\circ} n, θ = 138.18968 \dots^{\circ} + 36 0^{\circ} n, θ = - 41.81031 \dots^{\circ} + 36 0^{\circ} n, θ = 221.81031 \dots^{\circ} + 36 0^{\circ} n

求解步骤

9 cos (2 θ) = 9 cos^{2} (θ) - 4

两边减去

9 cos^{2} (θ) - 4

9 cos (2 θ) - 9 cos^{2} (θ) + 4 = 0

使用三角恒等式改写

4 + 9 cos (2 θ) - 9 cos^{2} (θ)

使用倍角公式:

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

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

= 4 + 9 cos (2 θ) - 9 (cos (2 θ) + sin^{2} (θ))

化简

4 + 9 cos (2 θ) - 9 (cos (2 θ) + sin^{2} (θ)) : 4 - 9 sin^{2} (θ)

4 + 9 cos (2 θ) - 9 (cos (2 θ) + sin^{2} (θ))

乘开

- 9 (cos (2 θ) + sin^{2} (θ)) : - 9 cos (2 θ) - 9 sin^{2} (θ)

- 9 (cos (2 θ) + sin^{2} (θ))

使用分配律:

a (b + c) = ab + a c

a = - 9, b = cos (2 θ), c = sin^{2} (θ)

= - 9 cos (2 θ) + (- 9) sin^{2} (θ)

使用加减运算法则

+ (- a) = - a

= - 9 cos (2 θ) - 9 sin^{2} (θ)

= 4 + 9 cos (2 θ) - 9 cos (2 θ) - 9 sin^{2} (θ)

同类项相加:

9 cos (2 θ) - 9 cos (2 θ) = 0

= 4 - 9 sin^{2} (θ)

= 4 - 9 sin^{2} (θ)

4 - 9 sin^{2} (θ) = 0

用替代法求解

4 - 9 sin^{2} (θ) = 0

令

: sin (θ) = u

4 - 9 u^{2} = 0

4 - 9 u^{2} = 0 : u = \frac{2}{3}, u = - \frac{2}{3}

4 - 9 u^{2} = 0

将

4

到右边

4 - 9 u^{2} = 0

两边减去

4

4 - 9 u^{2} - 4 = 0 - 4

化简

- 9 u^{2} = - 4

- 9 u^{2} = - 4

两边除以

- 9

- 9 u^{2} = - 4

两边除以

- 9

\frac{- 9 u ^{2}}{- 9} = \frac{- 4}{- 9}

化简

u^{2} = \frac{4}{9}

u^{2} = \frac{4}{9}

对于

x^{2} = f (a)

解为

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

u = \frac{4}{9}, u = - \frac{4}{9}

\frac{4}{9} = \frac{2}{3}

\frac{4}{9}

使用根式运算法则:

假定

a \geq 0, b \geq 0

= \frac{4}{9}

9 = 3

9

因式分解数字:

9 = 3^{2}

= 3^{2}

使用根式运算法则:

3^{2} = 3

= 3

= \frac{4}{3}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

2^{2} = 2

= 2

= \frac{2}{3}

- \frac{4}{9} = - \frac{2}{3}

- \frac{4}{9}

化简

\frac{4}{9} : \frac{2}{3}

\frac{4}{9}

使用根式运算法则:

假定

a \geq 0, b \geq 0

= \frac{4}{9}

9 = 3

9

因式分解数字:

9 = 3^{2}

= 3^{2}

使用根式运算法则:

3^{2} = 3

= 3

= \frac{4}{3}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

2^{2} = 2

= 2

= \frac{2}{3}

= - \frac{2}{3}

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

u = sin (θ)

代回

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

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

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

sin (θ) = \frac{2}{3}

使用反三角函数性质

sin (θ) = \frac{2}{3}

sin (θ) = \frac{2}{3}

的通解

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

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

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

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

sin (θ) = - \frac{2}{3}

使用反三角函数性质

sin (θ) = - \frac{2}{3}

sin (θ) = - \frac{2}{3}

的通解

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

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

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

合并所有解

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

以小数形式表示解

θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn, θ = - 0.72972 \dots + 2 πn, θ = π + 0.72972 \dots + 2 πn

作图

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