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受欢迎的三角函数 >

-3sqrt(3)sin(v)+3cos(v)=3sqrt(2)

解答

- 33 sin (v) + 3 cos (v) = 32

解答

v = π + 1.30899 \dots + 2 πn, v = - 0.26179 \dots + 2 πn

+1

度数

v = 25 5^{\circ} + 36 0^{\circ} n, v = - 1 5^{\circ} + 36 0^{\circ} n

求解步骤

- 33 sin (v) + 3 cos (v) = 32

两边加上

33 sin (v)

3 cos (v) = 32 + 33 sin (v)

两边进行平方

(3 cos (v))^{2} = (32 + 33 sin (v))^{2}

两边减去

(32 + 33 sin (v))^{2}

9 cos^{2} (v) - 18 - 186 sin (v) - 27 sin^{2} (v) = 0

使用三角恒等式改写

- 18 - 27 sin^{2} (v) + 9 cos^{2} (v) - 18 sin (v) 6

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

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

= - 18 - 27 sin^{2} (v) + 9 (1 - sin^{2} (v)) - 18 sin (v) 6

化简

- 18 - 27 sin^{2} (v) + 9 (1 - sin^{2} (v)) - 18 sin (v) 6 : - 36 sin^{2} (v) - 186 sin (v) - 9

- 18 - 27 sin^{2} (v) + 9 (1 - sin^{2} (v)) - 18 sin (v) 6

= - 18 - 27 sin^{2} (v) + 9 (1 - sin^{2} (v)) - 186 sin (v)

乘开

9 (1 - sin^{2} (v)) : 9 - 9 sin^{2} (v)

9 (1 - sin^{2} (v))

使用分配律:

a (b - c) = ab - a c

a = 9, b = 1, c = sin^{2} (v)

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

数字相乘:

9 \cdot 1 = 9

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

= - 18 - 27 sin^{2} (v) + 9 - 9 sin^{2} (v) - 18 sin (v) 6

化简

- 18 - 27 sin^{2} (v) + 9 - 9 sin^{2} (v) - 18 sin (v) 6 : - 36 sin^{2} (v) - 186 sin (v) - 9

- 18 - 27 sin^{2} (v) + 9 - 9 sin^{2} (v) - 18 sin (v) 6

对同类项分组

= - 27 sin^{2} (v) - 9 sin^{2} (v) - 186 sin (v) - 18 + 9

同类项相加:

- 27 sin^{2} (v) - 9 sin^{2} (v) = - 36 sin^{2} (v)

= - 36 sin^{2} (v) - 186 sin (v) - 18 + 9

数字相加/相减:

- 18 + 9 = - 9

= - 36 sin^{2} (v) - 186 sin (v) - 9

= - 36 sin^{2} (v) - 186 sin (v) - 9

= - 36 sin^{2} (v) - 186 sin (v) - 9

- 9 - 36 sin^{2} (v) - 18 sin (v) 6 = 0

用替代法求解

- 9 - 36 sin^{2} (v) - 18 sin (v) 6 = 0

令

: sin (v) = u

- 9 - 36 u^{2} - 18 u 6 = 0

- 9 - 36 u^{2} - 18 u 6 = 0 : u = - \frac{6 + 2}{4}, u = - \frac{6 - 2}{4}

- 9 - 36 u^{2} - 18 u 6 = 0

改写成标准形式

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

- 36 u^{2} - 186 u - 9 = 0

使用求根公式求解

- 36 u^{2} - 186 u - 9 = 0

二次方程求根公式:

若

a = - 36, b = - 186, c = - 9

u_{1, 2} = \frac{- ( - 18 6 ) \pm ( - 18 6 ) ^{2} - 4 ( - 36 ) ( - 9 )}{2 ( - 36 )}

u_{1, 2} = \frac{- ( - 18 6 ) \pm ( - 18 6 ) ^{2} - 4 ( - 36 ) ( - 9 )}{2 ( - 36 )}

(- 186)^{2} - 4 (- 36) (- 9) = 182

(- 186)^{2} - 4 (- 36) (- 9)

使用法则

- (- a) = a

= (- 186)^{2} - 4 \cdot 36 \cdot 9

(- 186)^{2} = 1 8^{2} \cdot 6

(- 186)^{2}

使用指数法则:

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

若

n

是偶数

(- 186)^{2} = (186)^{2}

= (186)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 1 8^{2} (6)^{2}

(6)^{2} : 6

使用根式运算法则:

a = a^{\frac{1}{2}}

= (6^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 6^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 6

= 1 8^{2} \cdot 6

4 \cdot 36 \cdot 9 = 1296

4 \cdot 36 \cdot 9

数字相乘:

4 \cdot 36 \cdot 9 = 1296

= 1296

= 1 8^{2} \cdot 6 - 1296

1 8^{2} \cdot 6 = 1944

1 8^{2} \cdot 6

1 8^{2} = 324

= 324 \cdot 6

数字相乘:

324 \cdot 6 = 1944

= 1944

= 1944 - 1296

数字相减:

1944 - 1296 = 648

= 648

648

质因数分解:

2^{3} \cdot 3^{4}

648

648

除以

2648 = 324 \cdot 2

= 2 \cdot 324

324

除以

2324 = 162 \cdot 2

= 2 \cdot 2 \cdot 162

162

除以

2162 = 81 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 81

81

除以

381 = 27 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 27

27

除以

327 = 9 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 9

9

除以

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3

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

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

使用指数法则:

a^{b + c} = a^{b} \cdot a^{c}

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

使用根式运算法则:

n ab = n a n b

= 2 2^{2} 3^{4}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 22 3^{4}

使用根式运算法则:

n a^{m} = a^{\frac{m}{n}}

3^{4} = 3^{\frac{4}{2}} = 3^{2}

= 3^{2} \cdot 22

整理后得

= 182

u_{1, 2} = \frac{- ( - 18 6 ) \pm 18 2}{2 ( - 36 )}

将解分隔开

u_{1} = \frac{- ( - 18 6 ) + 18 2}{2 ( - 36 )}, u_{2} = \frac{- ( - 18 6 ) - 18 2}{2 ( - 36 )}

u = \frac{- ( - 18 6 ) + 18 2}{2 ( - 36 )} : - \frac{6 + 2}{4}

\frac{- ( - 18 6 ) + 18 2}{2 ( - 36 )}

去除括号:

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

= \frac{18 6 + 18 2}{- 2 \cdot 36}

数字相乘:

2 \cdot 36 = 72

= \frac{18 6 + 18 2}{- 72}

使用分式法则:

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

= - \frac{18 6 + 18 2}{72}

消掉

\frac{18 6 + 18 2}{72} : \frac{6 + 2}{4}

\frac{18 6 + 18 2}{72}

因式分解出通项

18

= \frac{18 ( 6 + 2 )}{72}

约分:

18

= \frac{6 + 2}{4}

= - \frac{6 + 2}{4}

u = \frac{- ( - 18 6 ) - 18 2}{2 ( - 36 )} : - \frac{6 - 2}{4}

\frac{- ( - 18 6 ) - 18 2}{2 ( - 36 )}

去除括号:

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

= \frac{18 6 - 18 2}{- 2 \cdot 36}

数字相乘:

2 \cdot 36 = 72

= \frac{18 6 - 18 2}{- 72}

使用分式法则:

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

= - \frac{18 6 - 18 2}{72}

消掉

\frac{18 6 - 18 2}{72} : \frac{6 - 2}{4}

\frac{18 6 - 18 2}{72}

因式分解出通项

18

= \frac{18 ( 6 - 2 )}{72}

约分:

18

= \frac{6 - 2}{4}

= - \frac{6 - 2}{4}

二次方程组的解是:

u = - \frac{6 + 2}{4}, u = - \frac{6 - 2}{4}

u = sin (v)

代回

sin (v) = - \frac{6 + 2}{4}, sin (v) = - \frac{6 - 2}{4}

sin (v) = - \frac{6 + 2}{4}, sin (v) = - \frac{6 - 2}{4}

sin (v) = - \frac{6 + 2}{4} : v = arcsin (- \frac{6 + 2}{4}) + 2 πn, v = π + arcsin (\frac{6 + 2}{4}) + 2 πn

sin (v) = - \frac{6 + 2}{4}

使用反三角函数性质

sin (v) = - \frac{6 + 2}{4}

sin (v) = - \frac{6 + 2}{4}

的通解

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

v = arcsin (- \frac{6 + 2}{4}) + 2 πn, v = π + arcsin (\frac{6 + 2}{4}) + 2 πn

v = arcsin (- \frac{6 + 2}{4}) + 2 πn, v = π + arcsin (\frac{6 + 2}{4}) + 2 πn

sin (v) = - \frac{6 - 2}{4} : v = arcsin (- \frac{6 - 2}{4}) + 2 πn, v = π + arcsin (\frac{6 - 2}{4}) + 2 πn

sin (v) = - \frac{6 - 2}{4}

使用反三角函数性质

sin (v) = - \frac{6 - 2}{4}

sin (v) = - \frac{6 - 2}{4}

的通解

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

v = arcsin (- \frac{6 - 2}{4}) + 2 πn, v = π + arcsin (\frac{6 - 2}{4}) + 2 πn

v = arcsin (- \frac{6 - 2}{4}) + 2 πn, v = π + arcsin (\frac{6 - 2}{4}) + 2 πn

合并所有解

v = arcsin (- \frac{6 + 2}{4}) + 2 πn, v = π + arcsin (\frac{6 + 2}{4}) + 2 πn, v = arcsin (- \frac{6 - 2}{4}) + 2 πn, v = π + arcsin (\frac{6 - 2}{4}) + 2 πn

将解代入原方程进行验证

将它们代入

- 33 sin (v) + 3 cos (v) = 32

检验解是否符合

去除与方程不符的解。

检验

arcsin (- \frac{6 + 2}{4}) + 2 πn

的解:

假

arcsin (- \frac{6 + 2}{4}) + 2 πn

代入

n = 1

arcsin (- \frac{6 + 2}{4}) + 2 π 1

对于

- 33 sin (v) + 3 cos (v) = 32

代入

v = arcsin (- \frac{6 + 2}{4}) + 2 π 1

- 33 sin (arcsin (- \frac{6 + 2}{4}) + 2 π 1) + 3 cos (arcsin (- \frac{6 + 2}{4}) + 2 π 1) = 32

整理后得

5.79555 \dots = 4.24264 \dots

\Rightarrow 假

检验

π + arcsin (\frac{6 + 2}{4}) + 2 πn

的解:

真

π + arcsin (\frac{6 + 2}{4}) + 2 πn

代入

n = 1

π + arcsin (\frac{6 + 2}{4}) + 2 π 1

对于

- 33 sin (v) + 3 cos (v) = 32

代入

v = π + arcsin (\frac{6 + 2}{4}) + 2 π 1

- 33 sin (π + arcsin (\frac{6 + 2}{4}) + 2 π 1) + 3 cos (π + arcsin (\frac{6 + 2}{4}) + 2 π 1) = 32

整理后得

4.24264 \dots = 4.24264 \dots

\Rightarrow 真

检验

arcsin (- \frac{6 - 2}{4}) + 2 πn

的解:

真

arcsin (- \frac{6 - 2}{4}) + 2 πn

代入

n = 1

arcsin (- \frac{6 - 2}{4}) + 2 π 1

对于

- 33 sin (v) + 3 cos (v) = 32

代入

v = arcsin (- \frac{6 - 2}{4}) + 2 π 1

- 33 sin (arcsin (- \frac{6 - 2}{4}) + 2 π 1) + 3 cos (arcsin (- \frac{6 - 2}{4}) + 2 π 1) = 32

整理后得

4.24264 \dots = 4.24264 \dots

\Rightarrow 真

检验

π + arcsin (\frac{6 - 2}{4}) + 2 πn

的解:

假

π + arcsin (\frac{6 - 2}{4}) + 2 πn

代入

n = 1

π + arcsin (\frac{6 - 2}{4}) + 2 π 1

对于

- 33 sin (v) + 3 cos (v) = 32

代入

v = π + arcsin (\frac{6 - 2}{4}) + 2 π 1

- 33 sin (π + arcsin (\frac{6 - 2}{4}) + 2 π 1) + 3 cos (π + arcsin (\frac{6 - 2}{4}) + 2 π 1) = 32

整理后得

- 1.55291 \dots = 4.24264 \dots

\Rightarrow 假

v = π + arcsin (\frac{6 + 2}{4}) + 2 πn, v = arcsin (- \frac{6 - 2}{4}) + 2 πn

以小数形式表示解

v = π + 1.30899 \dots + 2 πn, v = - 0.26179 \dots + 2 πn

作图

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