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受欢迎的三角函数 >

6sin(x)+cos^2(x)=2

解答

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

解答

x = 0.17242 \dots + 2 πn, x = π - 0.17242 \dots + 2 πn

+1

度数

x = 9.87928 \dots^{\circ} + 36 0^{\circ} n, x = 170.12071 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

2

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

使用三角恒等式改写

- 2 + cos^{2} (x) + 6 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)

= - 2 + 1 - sin^{2} (x) + 6 sin (x)

化简

= 6 sin (x) - sin^{2} (x) - 1

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

用替代法求解

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

令

: sin (x) = u

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

- 1 - u^{2} + 6 u = 0 : u = 3 - 22, u = 3 + 22

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

改写成标准形式

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

- u^{2} + 6 u - 1 = 0

使用求根公式求解

- u^{2} + 6 u - 1 = 0

二次方程求根公式:

若

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

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

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

6^{2} - 4 (- 1) (- 1) = 42

6^{2} - 4 (- 1) (- 1)

使用法则

- (- a) = a

= 6^{2} - 4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 6^{2} - 4

6^{2} = 36

= 36 - 4

数字相减:

36 - 4 = 32

= 32

32

质因数分解:

2^{5}

32

32

除以

232 = 16 \cdot 2

= 2 \cdot 16

16

除以

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

除以

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

除以

24 = 2 \cdot 2

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

2

是质数，因此无法进一步因数分解

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

= 2^{5}

= 2^{5}

使用指数法则:

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

= 2^{4} \cdot 2

使用根式运算法则:

n ab = n a n b

= 2 2^{4}

使用根式运算法则:

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

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

= 2^{2} 2

整理后得

= 42

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

将解分隔开

u_{1} = \frac{- 6 + 4 2}{2 ( - 1 )}, u_{2} = \frac{- 6 - 4 2}{2 ( - 1 )}

u = \frac{- 6 + 4 2}{2 ( - 1 )} : 3 - 22

\frac{- 6 + 4 2}{2 ( - 1 )}

去除括号:

(- a) = - a

= \frac{- 6 + 4 2}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

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

使用分式法则:

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

- 6 + 42 = - (6 - 42)

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

分解

6 - 42 : 2 (3 - 22)

6 - 42

改写为

= 2 \cdot 3 - 2 \cdot 22

因式分解出通项

2

= 2 (3 - 22)

= \frac{2 ( 3 - 2 2 )}{2}

数字相除:

\frac{2}{2} = 1

= 3 - 22

u = \frac{- 6 - 4 2}{2 ( - 1 )} : 3 + 22

\frac{- 6 - 4 2}{2 ( - 1 )}

去除括号:

(- a) = - a

= \frac{- 6 - 4 2}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

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

使用分式法则:

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

- 6 - 42 = - (6 + 42)

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

分解

6 + 42 : 2 (3 + 22)

6 + 42

改写为

= 2 \cdot 3 + 2 \cdot 22

因式分解出通项

2

= 2 (3 + 22)

= \frac{2 ( 3 + 2 2 )}{2}

数字相除:

\frac{2}{2} = 1

= 3 + 22

二次方程组的解是:

u = 3 - 22, u = 3 + 22

u = sin (x)

代回

sin (x) = 3 - 22, sin (x) = 3 + 22

sin (x) = 3 - 22, sin (x) = 3 + 22

sin (x) = 3 - 22 : x = arcsin (3 - 22) + 2 πn, x = π - arcsin (3 - 22) + 2 πn

sin (x) = 3 - 22

使用反三角函数性质

sin (x) = 3 - 22

sin (x) = 3 - 22

的通解

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

x = arcsin (3 - 22) + 2 πn, x = π - arcsin (3 - 22) + 2 πn

x = arcsin (3 - 22) + 2 πn, x = π - arcsin (3 - 22) + 2 πn

sin (x) = 3 + 22 :

无解

sin (x) = 3 + 22

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = arcsin (3 - 22) + 2 πn, x = π - arcsin (3 - 22) + 2 πn

以小数形式表示解

x = 0.17242 \dots + 2 πn, x = π - 0.17242 \dots + 2 πn

作图

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