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受欢迎的三角函数 >

4cos^2(x)+17sin(x)=8

解答

4 cos^{2} (x) + 17 sin (x) = 8

解答

x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn

+1

度数

x = 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 165.52248 \dots^{\circ} + 36 0^{\circ} n

求解步骤

4 cos^{2} (x) + 17 sin (x) = 8

两边减去

8

4 cos^{2} (x) + 17 sin (x) - 8 = 0

使用三角恒等式改写

- 8 + 17 sin (x) + 4 cos^{2} (x)

使用毕达哥拉斯恒等式:

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

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

= - 8 + 17 sin (x) + 4 (1 - sin^{2} (x))

化简

- 8 + 17 sin (x) + 4 (1 - sin^{2} (x)) : 17 sin (x) - 4 sin^{2} (x) - 4

- 8 + 17 sin (x) + 4 (1 - sin^{2} (x))

乘开

4 (1 - sin^{2} (x)) : 4 - 4 sin^{2} (x)

4 (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 4, b = 1, c = sin^{2} (x)

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

数字相乘:

4 \cdot 1 = 4

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

= - 8 + 17 sin (x) + 4 - 4 sin^{2} (x)

化简

- 8 + 17 sin (x) + 4 - 4 sin^{2} (x) : 17 sin (x) - 4 sin^{2} (x) - 4

- 8 + 17 sin (x) + 4 - 4 sin^{2} (x)

对同类项分组

= 17 sin (x) - 4 sin^{2} (x) - 8 + 4

数字相加/相减:

- 8 + 4 = - 4

= 17 sin (x) - 4 sin^{2} (x) - 4

= 17 sin (x) - 4 sin^{2} (x) - 4

= 17 sin (x) - 4 sin^{2} (x) - 4

- 4 + 17 sin (x) - 4 sin^{2} (x) = 0

用替代法求解

- 4 + 17 sin (x) - 4 sin^{2} (x) = 0

令

: sin (x) = u

- 4 + 17 u - 4 u^{2} = 0

- 4 + 17 u - 4 u^{2} = 0 : u = \frac{1}{4}, u = 4

- 4 + 17 u - 4 u^{2} = 0

改写成标准形式

a x^{2} + b x + c = 0

- 4 u^{2} + 17 u - 4 = 0

使用求根公式求解

- 4 u^{2} + 17 u - 4 = 0

二次方程求根公式:

若

a = - 4, b = 17, c = - 4

u_{1, 2} = \frac{- 17 \pm 1 7 ^{2} - 4 ( - 4 ) ( - 4 )}{2 ( - 4 )}

u_{1, 2} = \frac{- 17 \pm 1 7 ^{2} - 4 ( - 4 ) ( - 4 )}{2 ( - 4 )}

1 7^{2} - 4 (- 4) (- 4) = 15

1 7^{2} - 4 (- 4) (- 4)

使用法则

- (- a) = a

= 1 7^{2} - 4 \cdot 4 \cdot 4

数字相乘:

4 \cdot 4 \cdot 4 = 64

= 1 7^{2} - 64

1 7^{2} = 289

= 289 - 64

数字相减:

289 - 64 = 225

= 225

因式分解数字:

225 = 1 5^{2}

= 1 5^{2}

使用根式运算法则:

1 5^{2} = 15

= 15

u_{1, 2} = \frac{- 17 \pm 15}{2 ( - 4 )}

将解分隔开

u_{1} = \frac{- 17 + 15}{2 ( - 4 )}, u_{2} = \frac{- 17 - 15}{2 ( - 4 )}

u = \frac{- 17 + 15}{2 ( - 4 )} : \frac{1}{4}

\frac{- 17 + 15}{2 ( - 4 )}

去除括号:

(- a) = - a

= \frac{- 17 + 15}{- 2 \cdot 4}

数字相加/相减:

- 17 + 15 = - 2

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

数字相乘:

2 \cdot 4 = 8

= \frac{- 2}{- 8}

使用分式法则:

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

= \frac{2}{8}

约分:

2

= \frac{1}{4}

u = \frac{- 17 - 15}{2 ( - 4 )} : 4

\frac{- 17 - 15}{2 ( - 4 )}

去除括号:

(- a) = - a

= \frac{- 17 - 15}{- 2 \cdot 4}

数字相减:

- 17 - 15 = - 32

= \frac{- 32}{- 2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 32}{- 8}

使用分式法则:

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

= \frac{32}{8}

数字相除:

\frac{32}{8} = 4

= 4

二次方程组的解是:

u = \frac{1}{4}, u = 4

u = sin (x)

代回

sin (x) = \frac{1}{4}, sin (x) = 4

sin (x) = \frac{1}{4}, sin (x) = 4

sin (x) = \frac{1}{4} : x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

sin (x) = \frac{1}{4}

使用反三角函数性质

sin (x) = \frac{1}{4}

sin (x) = \frac{1}{4}

的通解

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

x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

sin (x) = 4 :

无解

sin (x) = 4

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

以小数形式表示解

x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn

作图

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