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受欢迎的三角函数 >

3sin(x)+cos^2(x)=2

解答

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

解答

x = 0.39192 \dots + 2 πn, x = π - 0.39192 \dots + 2 πn

+1

度数

x = 22.45551 \dots^{\circ} + 36 0^{\circ} n, x = 157.54448 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

2

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

使用三角恒等式改写

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

化简

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

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

用替代法求解

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

令

: sin (x) = u

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

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

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

改写成标准形式

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

- u^{2} + 3 u - 1 = 0

使用求根公式求解

- u^{2} + 3 u - 1 = 0

二次方程求根公式:

若

a = - 1, b = 3, c = - 1

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

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

3^{2} - 4 (- 1) (- 1) = 5

3^{2} - 4 (- 1) (- 1)

使用法则

- (- a) = a

= 3^{2} - 4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 3^{2} - 4

3^{2} = 9

= 9 - 4

数字相减:

9 - 4 = 5

= 5

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

将解分隔开

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

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

\frac{- 3 + 5}{2 ( - 1 )}

去除括号:

(- a) = - a

= \frac{- 3 + 5}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

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

使用分式法则:

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

- 3 + 5 = - (3 - 5)

= \frac{3 - 5}{2}

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

\frac{- 3 - 5}{2 ( - 1 )}

去除括号:

(- a) = - a

= \frac{- 3 - 5}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

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

使用分式法则:

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

- 3 - 5 = - (3 + 5)

= \frac{3 + 5}{2}

二次方程组的解是:

u = \frac{3 - 5}{2}, u = \frac{3 + 5}{2}

u = sin (x)

代回

sin (x) = \frac{3 - 5}{2}, sin (x) = \frac{3 + 5}{2}

sin (x) = \frac{3 - 5}{2}, sin (x) = \frac{3 + 5}{2}

sin (x) = \frac{3 - 5}{2} : x = arcsin (\frac{3 - 5}{2}) + 2 πn, x = π - arcsin (\frac{3 - 5}{2}) + 2 πn

sin (x) = \frac{3 - 5}{2}

使用反三角函数性质

sin (x) = \frac{3 - 5}{2}

sin (x) = \frac{3 - 5}{2}

的通解

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

x = arcsin (\frac{3 - 5}{2}) + 2 πn, x = π - arcsin (\frac{3 - 5}{2}) + 2 πn

x = arcsin (\frac{3 - 5}{2}) + 2 πn, x = π - arcsin (\frac{3 - 5}{2}) + 2 πn

sin (x) = \frac{3 + 5}{2} :

无解

sin (x) = \frac{3 + 5}{2}

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = arcsin (\frac{3 - 5}{2}) + 2 πn, x = π - arcsin (\frac{3 - 5}{2}) + 2 πn

以小数形式表示解

x = 0.39192 \dots + 2 πn, x = π - 0.39192 \dots + 2 πn

作图

流行的例子

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5 sin (2 θ) - 7 sin (θ) = 0

cos(θ)=(sqrt(3))/3

cos (θ) = \frac{3}{3}

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