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受欢迎的三角函数 >

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

解答

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

解答

x = - 0.41151 \dots + 2 πn, x = π + 0.41151 \dots + 2 πn, x = \frac{π}{2} + 2 πn

+1

度数

x = - 23.57817 \dots^{\circ} + 36 0^{\circ} n, x = 203.57817 \dots^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

- 3 + 3 sin (x) + 5 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)

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

化简

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

a = 5, b = 1, c = sin^{2} (x)

= 5 \cdot 1 - 5 sin^{2} (x)

数字相乘:

5 \cdot 1 = 5

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

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

化简

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

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

对同类项分组

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

数字相加/相减:

- 3 + 5 = 2

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

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

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

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

用替代法求解

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

令

: sin (x) = u

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

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

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

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

使用法则

- (- a) = a

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

数字相乘:

4 \cdot 5 \cdot 2 = 40

= 3^{2} + 40

3^{2} = 9

= 9 + 40

数字相加:

9 + 40 = 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{- 3 \pm 7}{2 ( - 5 )}

将解分隔开

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

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

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

去除括号:

(- a) = - a

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

数字相加/相减:

- 3 + 7 = 4

= \frac{4}{- 2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{4}{- 10}

使用分式法则:

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

= - \frac{4}{10}

约分:

2

= - \frac{2}{5}

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

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

去除括号:

(- a) = - a

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

数字相减:

- 3 - 7 = - 10

= \frac{- 10}{- 2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{- 10}{- 10}

使用分式法则:

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

= \frac{10}{10}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

u = - \frac{2}{5}, u = 1

u = sin (x)

代回

sin (x) = - \frac{2}{5}, sin (x) = 1

sin (x) = - \frac{2}{5}, sin (x) = 1

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

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

使用反三角函数性质

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

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

的通解

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

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

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

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

sin (x) = 1

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

合并所有解

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

以小数形式表示解

x = - 0.41151 \dots + 2 πn, x = π + 0.41151 \dots + 2 πn, x = \frac{π}{2} + 2 πn

作图

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