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受欢迎的三角函数 >

15cos^2(x)+sin(x)-9=0

解答

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

解答

x = - 0.64350 \dots + 2 πn, x = π + 0.64350 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

+1

度数

x = - 36.86989 \dots^{\circ} + 36 0^{\circ} n, x = 216.86989 \dots^{\circ} + 36 0^{\circ} n, x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

- 9 + sin (x) + 15 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)

= - 9 + sin (x) + 15 (1 - sin^{2} (x))

化简

- 9 + sin (x) + 15 (1 - sin^{2} (x)) : sin (x) - 15 sin^{2} (x) + 6

- 9 + sin (x) + 15 (1 - sin^{2} (x))

乘开

15 (1 - sin^{2} (x)) : 15 - 15 sin^{2} (x)

15 (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 15, b = 1, c = sin^{2} (x)

= 15 \cdot 1 - 15 sin^{2} (x)

数字相乘:

15 \cdot 1 = 15

= 15 - 15 sin^{2} (x)

= - 9 + sin (x) + 15 - 15 sin^{2} (x)

化简

- 9 + sin (x) + 15 - 15 sin^{2} (x) : sin (x) - 15 sin^{2} (x) + 6

- 9 + sin (x) + 15 - 15 sin^{2} (x)

对同类项分组

= sin (x) - 15 sin^{2} (x) - 9 + 15

数字相加/相减:

- 9 + 15 = 6

= sin (x) - 15 sin^{2} (x) + 6

= sin (x) - 15 sin^{2} (x) + 6

= sin (x) - 15 sin^{2} (x) + 6

6 + sin (x) - 15 sin^{2} (x) = 0

用替代法求解

6 + sin (x) - 15 sin^{2} (x) = 0

令

: sin (x) = u

6 + u - 15 u^{2} = 0

6 + u - 15 u^{2} = 0 : u = - \frac{3}{5}, u = \frac{2}{3}

6 + u - 15 u^{2} = 0

改写成标准形式

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

- 15 u^{2} + u + 6 = 0

使用求根公式求解

- 15 u^{2} + u + 6 = 0

二次方程求根公式:

若

a = - 15, b = 1, c = 6

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 15 ) \cdot 6}{2 ( - 15 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 15 ) \cdot 6}{2 ( - 15 )}

1^{2} - 4 (- 15) \cdot 6 = 19

1^{2} - 4 (- 15) \cdot 6

使用法则

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 15) \cdot 6

使用法则

- (- a) = a

= 1 + 4 \cdot 15 \cdot 6

数字相乘:

4 \cdot 15 \cdot 6 = 360

= 1 + 360

数字相加:

1 + 360 = 361

= 361

因式分解数字:

361 = 1 9^{2}

= 1 9^{2}

使用根式运算法则:

n a^{n} = a

1 9^{2} = 19

= 19

u_{1, 2} = \frac{- 1 \pm 19}{2 ( - 15 )}

将解分隔开

u_{1} = \frac{- 1 + 19}{2 ( - 15 )}, u_{2} = \frac{- 1 - 19}{2 ( - 15 )}

u = \frac{- 1 + 19}{2 ( - 15 )} : - \frac{3}{5}

\frac{- 1 + 19}{2 ( - 15 )}

去除括号:

(- a) = - a

= \frac{- 1 + 19}{- 2 \cdot 15}

数字相加/相减:

- 1 + 19 = 18

= \frac{18}{- 2 \cdot 15}

数字相乘:

2 \cdot 15 = 30

= \frac{18}{- 30}

使用分式法则:

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

= - \frac{18}{30}

约分:

6

= - \frac{3}{5}

u = \frac{- 1 - 19}{2 ( - 15 )} : \frac{2}{3}

\frac{- 1 - 19}{2 ( - 15 )}

去除括号:

(- a) = - a

= \frac{- 1 - 19}{- 2 \cdot 15}

数字相减:

- 1 - 19 = - 20

= \frac{- 20}{- 2 \cdot 15}

数字相乘:

2 \cdot 15 = 30

= \frac{- 20}{- 30}

使用分式法则:

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

= \frac{20}{30}

约分:

10

= \frac{2}{3}

二次方程组的解是:

u = - \frac{3}{5}, u = \frac{2}{3}

u = sin (x)

代回

sin (x) = - \frac{3}{5}, sin (x) = \frac{2}{3}

sin (x) = - \frac{3}{5}, sin (x) = \frac{2}{3}

sin (x) = - \frac{3}{5} : x = arcsin (- \frac{3}{5}) + 2 πn, x = π + arcsin (\frac{3}{5}) + 2 πn

sin (x) = - \frac{3}{5}

使用反三角函数性质

sin (x) = - \frac{3}{5}

sin (x) = - \frac{3}{5}

的通解

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

x = arcsin (- \frac{3}{5}) + 2 πn, x = π + arcsin (\frac{3}{5}) + 2 πn

x = arcsin (- \frac{3}{5}) + 2 πn, x = π + arcsin (\frac{3}{5}) + 2 πn

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

sin (x) = \frac{2}{3}

使用反三角函数性质

sin (x) = \frac{2}{3}

sin (x) = \frac{2}{3}

的通解

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

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

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

合并所有解

x = arcsin (- \frac{3}{5}) + 2 πn, x = π + arcsin (\frac{3}{5}) + 2 πn, x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

以小数形式表示解

x = - 0.64350 \dots + 2 πn, x = π + 0.64350 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

作图

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