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受欢迎的三角函数 >

2cos^2(x)-9sin(x)-3=0

解答

2 cos^{2} (x) - 9 sin (x) - 3 = 0

解答

x = - 0.11424 \dots + 2 πn, x = π + 0.11424 \dots + 2 πn

+1

度数

x = - 6.54589 \dots^{\circ} + 36 0^{\circ} n, x = 186.54589 \dots^{\circ} + 36 0^{\circ} n

求解步骤

2 cos^{2} (x) - 9 sin (x) - 3 = 0

使用三角恒等式改写

- 3 + 2 cos^{2} (x) - 9 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)

= - 3 + 2 (1 - sin^{2} (x)) - 9 sin (x)

化简

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

数字相乘:

2 \cdot 1 = 2

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

= - 3 + 2 - 2 sin^{2} (x) - 9 sin (x)

数字相加/相减:

- 3 + 2 = - 1

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

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

- 1 - 2 sin^{2} (x) - 9 sin (x) = 0

用替代法求解

- 1 - 2 sin^{2} (x) - 9 sin (x) = 0

令

: sin (x) = u

- 1 - 2 u^{2} - 9 u = 0

- 1 - 2 u^{2} - 9 u = 0 : u = - \frac{9 + 73}{4}, u = - \frac{9 - 73}{4}

- 1 - 2 u^{2} - 9 u = 0

改写成标准形式

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

- 2 u^{2} - 9 u - 1 = 0

使用求根公式求解

- 2 u^{2} - 9 u - 1 = 0

二次方程求根公式:

若

a = - 2, b = - 9, c = - 1

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 ( - 2 ) ( - 1 )}{2 ( - 2 )}

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 ( - 2 ) ( - 1 )}{2 ( - 2 )}

(- 9)^{2} - 4 (- 2) (- 1) = 73

(- 9)^{2} - 4 (- 2) (- 1)

使用法则

- (- a) = a

= (- 9)^{2} - 4 \cdot 2 \cdot 1

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 9)^{2} = 9^{2}

= 9^{2} - 4 \cdot 2 \cdot 1

数字相乘:

4 \cdot 2 \cdot 1 = 8

= 9^{2} - 8

9^{2} = 81

= 81 - 8

数字相减:

81 - 8 = 73

= 73

u_{1, 2} = \frac{- ( - 9 ) \pm 73}{2 ( - 2 )}

将解分隔开

u_{1} = \frac{- ( - 9 ) + 73}{2 ( - 2 )}, u_{2} = \frac{- ( - 9 ) - 73}{2 ( - 2 )}

u = \frac{- ( - 9 ) + 73}{2 ( - 2 )} : - \frac{9 + 73}{4}

\frac{- ( - 9 ) + 73}{2 ( - 2 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{9 + 73}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{9 + 73}{- 4}

使用分式法则:

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

= - \frac{9 + 73}{4}

u = \frac{- ( - 9 ) - 73}{2 ( - 2 )} : - \frac{9 - 73}{4}

\frac{- ( - 9 ) - 73}{2 ( - 2 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{9 - 73}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{9 - 73}{- 4}

使用分式法则:

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

= - \frac{9 - 73}{4}

二次方程组的解是:

u = - \frac{9 + 73}{4}, u = - \frac{9 - 73}{4}

u = sin (x)

代回

sin (x) = - \frac{9 + 73}{4}, sin (x) = - \frac{9 - 73}{4}

sin (x) = - \frac{9 + 73}{4}, sin (x) = - \frac{9 - 73}{4}

sin (x) = - \frac{9 + 73}{4} :

无解

sin (x) = - \frac{9 + 73}{4}

- 1 \leq sin (x) \leq 1

无解

sin (x) = - \frac{9 - 73}{4} : x = arcsin (- \frac{9 - 73}{4}) + 2 πn, x = π + arcsin (\frac{9 - 73}{4}) + 2 πn

sin (x) = - \frac{9 - 73}{4}

使用反三角函数性质

sin (x) = - \frac{9 - 73}{4}

sin (x) = - \frac{9 - 73}{4}

的通解

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

x = arcsin (- \frac{9 - 73}{4}) + 2 πn, x = π + arcsin (\frac{9 - 73}{4}) + 2 πn

x = arcsin (- \frac{9 - 73}{4}) + 2 πn, x = π + arcsin (\frac{9 - 73}{4}) + 2 πn

合并所有解

x = arcsin (- \frac{9 - 73}{4}) + 2 πn, x = π + arcsin (\frac{9 - 73}{4}) + 2 πn

以小数形式表示解

x = - 0.11424 \dots + 2 πn, x = π + 0.11424 \dots + 2 πn

作图

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