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受欢迎的三角函数 >

3cos^2(x)=1-5sin(x)

解答

3 cos^{2} (x) = 1 - 5 sin (x)

解答

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

+1

度数

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

求解步骤

3 cos^{2} (x) = 1 - 5 sin (x)

两边减去

1 - 5 sin (x)

3 cos^{2} (x) - 1 + 5 sin (x) = 0

使用三角恒等式改写

- 1 + 3 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)

= - 1 + 3 (1 - sin^{2} (x)) + 5 sin (x)

化简

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

a = 3, b = 1, c = sin^{2} (x)

= 3 \cdot 1 - 3 sin^{2} (x)

数字相乘:

3 \cdot 1 = 3

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

= - 1 + 3 - 3 sin^{2} (x) + 5 sin (x)

数字相加/相减:

- 1 + 3 = 2

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

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

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

用替代法求解

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

令

: sin (x) = u

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

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 3, b = 5, c = 2

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

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

5^{2} - 4 (- 3) \cdot 2 = 7

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

使用法则

- (- a) = a

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

数字相乘:

4 \cdot 3 \cdot 2 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

数字相加:

25 + 24 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- 5 \pm 7}{2 ( - 3 )}

将解分隔开

u_{1} = \frac{- 5 + 7}{2 ( - 3 )}, u_{2} = \frac{- 5 - 7}{2 ( - 3 )}

u = \frac{- 5 + 7}{2 ( - 3 )} : - \frac{1}{3}

\frac{- 5 + 7}{2 ( - 3 )}

去除括号:

(- a) = - a

= \frac{- 5 + 7}{- 2 \cdot 3}

数字相加/相减:

- 5 + 7 = 2

= \frac{2}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{2}{- 6}

使用分式法则:

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

= - \frac{2}{6}

约分:

2

= - \frac{1}{3}

u = \frac{- 5 - 7}{2 ( - 3 )} : 2

\frac{- 5 - 7}{2 ( - 3 )}

去除括号:

(- a) = - a

= \frac{- 5 - 7}{- 2 \cdot 3}

数字相减:

- 5 - 7 = - 12

= \frac{- 12}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 12}{- 6}

使用分式法则:

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

= \frac{12}{6}

数字相除:

\frac{12}{6} = 2

= 2

二次方程组的解是:

u = - \frac{1}{3}, u = 2

u = sin (x)

代回

sin (x) = - \frac{1}{3}, sin (x) = 2

sin (x) = - \frac{1}{3}, sin (x) = 2

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) = 2 :

无解

sin (x) = 2

- 1 \leq sin (x) \leq 1

无解

合并所有解

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

以小数形式表示解

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

作图

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