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受欢迎的三角函数 >

12cos^2(x)+5sin(x)-9=0

解答

12 cos^{2} (x) + 5 sin (x) - 9 = 0

解答

x = - 0.33983 \dots + 2 πn, x = π + 0.33983 \dots + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

+1

度数

x = - 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 199.47122 \dots^{\circ} + 36 0^{\circ} n, x = 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 131.40962 \dots^{\circ} + 36 0^{\circ} n

求解步骤

12 cos^{2} (x) + 5 sin (x) - 9 = 0

使用三角恒等式改写

- 9 + 12 cos^{2} (x) + 5 sin (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)

= - 9 + 12 (1 - sin^{2} (x)) + 5 sin (x)

化简

- 9 + 12 (1 - sin^{2} (x)) + 5 sin (x) : 5 sin (x) - 12 sin^{2} (x) + 3

- 9 + 12 (1 - sin^{2} (x)) + 5 sin (x)

乘开

12 (1 - sin^{2} (x)) : 12 - 12 sin^{2} (x)

12 (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 12, b = 1, c = sin^{2} (x)

= 12 \cdot 1 - 12 sin^{2} (x)

数字相乘:

12 \cdot 1 = 12

= 12 - 12 sin^{2} (x)

= - 9 + 12 - 12 sin^{2} (x) + 5 sin (x)

数字相加/相减:

- 9 + 12 = 3

= 5 sin (x) - 12 sin^{2} (x) + 3

= 5 sin (x) - 12 sin^{2} (x) + 3

3 - 12 sin^{2} (x) + 5 sin (x) = 0

用替代法求解

3 - 12 sin^{2} (x) + 5 sin (x) = 0

令

: sin (x) = u

3 - 12 u^{2} + 5 u = 0

3 - 12 u^{2} + 5 u = 0 : u = - \frac{1}{3}, u = \frac{3}{4}

3 - 12 u^{2} + 5 u = 0

改写成标准形式

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

- 12 u^{2} + 5 u + 3 = 0

使用求根公式求解

- 12 u^{2} + 5 u + 3 = 0

二次方程求根公式:

若

a = - 12, b = 5, c = 3

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 12 ) \cdot 3}{2 ( - 12 )}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 12 ) \cdot 3}{2 ( - 12 )}

5^{2} - 4 (- 12) \cdot 3 = 13

5^{2} - 4 (- 12) \cdot 3

使用法则

- (- a) = a

= 5^{2} + 4 \cdot 12 \cdot 3

数字相乘:

4 \cdot 12 \cdot 3 = 144

= 5^{2} + 144

5^{2} = 25

= 25 + 144

数字相加:

25 + 144 = 169

= 169

因式分解数字:

169 = 1 3^{2}

= 1 3^{2}

使用根式运算法则:

n a^{n} = a

1 3^{2} = 13

= 13

u_{1, 2} = \frac{- 5 \pm 13}{2 ( - 12 )}

将解分隔开

u_{1} = \frac{- 5 + 13}{2 ( - 12 )}, u_{2} = \frac{- 5 - 13}{2 ( - 12 )}

u = \frac{- 5 + 13}{2 ( - 12 )} : - \frac{1}{3}

\frac{- 5 + 13}{2 ( - 12 )}

去除括号:

(- a) = - a

= \frac{- 5 + 13}{- 2 \cdot 12}

数字相加/相减:

- 5 + 13 = 8

= \frac{8}{- 2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{8}{- 24}

使用分式法则:

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

= - \frac{8}{24}

约分:

8

= - \frac{1}{3}

u = \frac{- 5 - 13}{2 ( - 12 )} : \frac{3}{4}

\frac{- 5 - 13}{2 ( - 12 )}

去除括号:

(- a) = - a

= \frac{- 5 - 13}{- 2 \cdot 12}

数字相减:

- 5 - 13 = - 18

= \frac{- 18}{- 2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{- 18}{- 24}

使用分式法则:

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

= \frac{18}{24}

约分:

6

= \frac{3}{4}

二次方程组的解是:

u = - \frac{1}{3}, u = \frac{3}{4}

u = sin (x)

代回

sin (x) = - \frac{1}{3}, sin (x) = \frac{3}{4}

sin (x) = - \frac{1}{3}, sin (x) = \frac{3}{4}

sin (x) = - \frac{1}{3} : x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

sin (x) = - \frac{1}{3}

使用反三角函数性质

sin (x) = - \frac{1}{3}

sin (x) = - \frac{1}{3}

的通解

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

x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn

sin (x) = \frac{3}{4} : x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

sin (x) = \frac{3}{4}

使用反三角函数性质

sin (x) = \frac{3}{4}

sin (x) = \frac{3}{4}

的通解

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

x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

合并所有解

x = arcsin (- \frac{1}{3}) + 2 πn, x = π + arcsin (\frac{1}{3}) + 2 πn, x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

以小数形式表示解

x = - 0.33983 \dots + 2 πn, x = π + 0.33983 \dots + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

作图

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