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受欢迎的三角函数 >

7cos(2x)=7sin^2(x)+5

解答

7 cos (2 x) = 7 sin^{2} (x) + 5

解答

x = 0.31372 \dots + 2 πn, x = π - 0.31372 \dots + 2 πn, x = - 0.31372 \dots + 2 πn, x = π + 0.31372 \dots + 2 πn

+1

度数

x = 17.97528 \dots^{\circ} + 36 0^{\circ} n, x = 162.02471 \dots^{\circ} + 36 0^{\circ} n, x = - 17.97528 \dots^{\circ} + 36 0^{\circ} n, x = 197.97528 \dots^{\circ} + 36 0^{\circ} n

求解步骤

7 cos (2 x) = 7 sin^{2} (x) + 5

两边减去

7 sin^{2} (x) + 5

7 cos (2 x) - 7 sin^{2} (x) - 5 = 0

使用三角恒等式改写

- 5 + 7 cos (2 x) - 7 sin^{2} (x)

使用倍角公式:

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

= - 5 + 7 (1 - 2 sin^{2} (x)) - 7 sin^{2} (x)

化简

- 5 + 7 (1 - 2 sin^{2} (x)) - 7 sin^{2} (x) : - 21 sin^{2} (x) + 2

- 5 + 7 (1 - 2 sin^{2} (x)) - 7 sin^{2} (x)

乘开

7 (1 - 2 sin^{2} (x)) : 7 - 14 sin^{2} (x)

7 (1 - 2 sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 7, b = 1, c = 2 sin^{2} (x)

= 7 \cdot 1 - 7 \cdot 2 sin^{2} (x)

化简

7 \cdot 1 - 7 \cdot 2 sin^{2} (x) : 7 - 14 sin^{2} (x)

7 \cdot 1 - 7 \cdot 2 sin^{2} (x)

数字相乘:

7 \cdot 1 = 7

= 7 - 7 \cdot 2 sin^{2} (x)

数字相乘:

7 \cdot 2 = 14

= 7 - 14 sin^{2} (x)

= 7 - 14 sin^{2} (x)

= - 5 + 7 - 14 sin^{2} (x) - 7 sin^{2} (x)

化简

- 5 + 7 - 14 sin^{2} (x) - 7 sin^{2} (x) : - 21 sin^{2} (x) + 2

- 5 + 7 - 14 sin^{2} (x) - 7 sin^{2} (x)

同类项相加:

- 14 sin^{2} (x) - 7 sin^{2} (x) = - 21 sin^{2} (x)

= - 5 + 7 - 21 sin^{2} (x)

数字相加/相减:

- 5 + 7 = 2

= - 21 sin^{2} (x) + 2

= - 21 sin^{2} (x) + 2

= - 21 sin^{2} (x) + 2

2 - 21 sin^{2} (x) = 0

用替代法求解

2 - 21 sin^{2} (x) = 0

令

: sin (x) = u

2 - 21 u^{2} = 0

2 - 21 u^{2} = 0 : u = \frac{2}{21}, u = - \frac{2}{21}

2 - 21 u^{2} = 0

将

2

到右边

2 - 21 u^{2} = 0

两边减去

2

2 - 21 u^{2} - 2 = 0 - 2

化简

- 21 u^{2} = - 2

- 21 u^{2} = - 2

两边除以

- 21

- 21 u^{2} = - 2

两边除以

- 21

\frac{- 21 u ^{2}}{- 21} = \frac{- 2}{- 21}

化简

u^{2} = \frac{2}{21}

u^{2} = \frac{2}{21}

对于

x^{2} = f (a)

解为

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

u = \frac{2}{21}, u = - \frac{2}{21}

u = sin (x)

代回

sin (x) = \frac{2}{21}, sin (x) = - \frac{2}{21}

sin (x) = \frac{2}{21}, sin (x) = - \frac{2}{21}

sin (x) = \frac{2}{21} : x = arcsin (\frac{2}{21}) + 2 πn, x = π - arcsin (\frac{2}{21}) + 2 πn

sin (x) = \frac{2}{21}

使用反三角函数性质

sin (x) = \frac{2}{21}

sin (x) = \frac{2}{21}

的通解

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

x = arcsin (\frac{2}{21}) + 2 πn, x = π - arcsin (\frac{2}{21}) + 2 πn

x = arcsin (\frac{2}{21}) + 2 πn, x = π - arcsin (\frac{2}{21}) + 2 πn

sin (x) = - \frac{2}{21} : x = arcsin (- \frac{2}{21}) + 2 πn, x = π + arcsin (\frac{2}{21}) + 2 πn

sin (x) = - \frac{2}{21}

使用反三角函数性质

sin (x) = - \frac{2}{21}

sin (x) = - \frac{2}{21}

的通解

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

x = arcsin (- \frac{2}{21}) + 2 πn, x = π + arcsin (\frac{2}{21}) + 2 πn

x = arcsin (- \frac{2}{21}) + 2 πn, x = π + arcsin (\frac{2}{21}) + 2 πn

合并所有解

x = arcsin (\frac{2}{21}) + 2 πn, x = π - arcsin (\frac{2}{21}) + 2 πn, x = arcsin (- \frac{2}{21}) + 2 πn, x = π + arcsin (\frac{2}{21}) + 2 πn

以小数形式表示解

x = 0.31372 \dots + 2 πn, x = π - 0.31372 \dots + 2 πn, x = - 0.31372 \dots + 2 πn, x = π + 0.31372 \dots + 2 πn

作图

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