zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

4-7sin(x)=cos^2(x)

解答

4 - 7 sin (x) = cos^{2} (x)

解答

x = 0.47644 \dots + 2 πn, x = π - 0.47644 \dots + 2 πn

+1

度数

x = 27.29801 \dots^{\circ} + 36 0^{\circ} n, x = 152.70198 \dots^{\circ} + 36 0^{\circ} n

求解步骤

4 - 7 sin (x) = cos^{2} (x)

两边减去

cos^{2} (x)

4 - 7 sin (x) - cos^{2} (x) = 0

使用三角恒等式改写

4 - cos^{2} (x) - 7 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)

= 4 - (1 - sin^{2} (x)) - 7 sin (x)

化简

4 - (1 - sin^{2} (x)) - 7 sin (x) : sin^{2} (x) - 7 sin (x) + 3

4 - (1 - sin^{2} (x)) - 7 sin (x)

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

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

打开括号

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

使用加减运算法则

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

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

= 4 - 1 + sin^{2} (x) - 7 sin (x)

数字相减:

4 - 1 = 3

= sin^{2} (x) - 7 sin (x) + 3

= sin^{2} (x) - 7 sin (x) + 3

3 + sin^{2} (x) - 7 sin (x) = 0

用替代法求解

3 + sin^{2} (x) - 7 sin (x) = 0

令

: sin (x) = u

3 + u^{2} - 7 u = 0

3 + u^{2} - 7 u = 0 : u = \frac{7 + 37}{2}, u = \frac{7 - 37}{2}

3 + u^{2} - 7 u = 0

改写成标准形式

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

u^{2} - 7 u + 3 = 0

使用求根公式求解

u^{2} - 7 u + 3 = 0

二次方程求根公式:

若

a = 1, b = - 7, c = 3

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 1 \cdot 3}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 1 \cdot 3}{2 \cdot 1}

(- 7)^{2} - 4 \cdot 1 \cdot 3 = 37

(- 7)^{2} - 4 \cdot 1 \cdot 3

使用指数法则:

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

若

n

是偶数

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 1 \cdot 3

数字相乘:

4 \cdot 1 \cdot 3 = 12

= 7^{2} - 12

7^{2} = 49

= 49 - 12

数字相减:

49 - 12 = 37

= 37

u_{1, 2} = \frac{- ( - 7 ) \pm 37}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- ( - 7 ) + 37}{2 \cdot 1}, u_{2} = \frac{- ( - 7 ) - 37}{2 \cdot 1}

u = \frac{- ( - 7 ) + 37}{2 \cdot 1} : \frac{7 + 37}{2}

\frac{- ( - 7 ) + 37}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{7 + 37}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{7 + 37}{2}

u = \frac{- ( - 7 ) - 37}{2 \cdot 1} : \frac{7 - 37}{2}

\frac{- ( - 7 ) - 37}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{7 - 37}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{7 - 37}{2}

二次方程组的解是:

u = \frac{7 + 37}{2}, u = \frac{7 - 37}{2}

u = sin (x)

代回

sin (x) = \frac{7 + 37}{2}, sin (x) = \frac{7 - 37}{2}

sin (x) = \frac{7 + 37}{2}, sin (x) = \frac{7 - 37}{2}

sin (x) = \frac{7 + 37}{2} :

无解

sin (x) = \frac{7 + 37}{2}

- 1 \leq sin (x) \leq 1

无解

sin (x) = \frac{7 - 37}{2} : x = arcsin (\frac{7 - 37}{2}) + 2 πn, x = π - arcsin (\frac{7 - 37}{2}) + 2 πn

sin (x) = \frac{7 - 37}{2}

使用反三角函数性质

sin (x) = \frac{7 - 37}{2}

sin (x) = \frac{7 - 37}{2}

的通解

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

x = arcsin (\frac{7 - 37}{2}) + 2 πn, x = π - arcsin (\frac{7 - 37}{2}) + 2 πn

x = arcsin (\frac{7 - 37}{2}) + 2 πn, x = π - arcsin (\frac{7 - 37}{2}) + 2 πn

合并所有解

x = arcsin (\frac{7 - 37}{2}) + 2 πn, x = π - arcsin (\frac{7 - 37}{2}) + 2 πn

以小数形式表示解

x = 0.47644 \dots + 2 πn, x = π - 0.47644 \dots + 2 πn

作图

流行的例子

(d^2-2d+3)=sin(x)

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

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

tan^{22}(x)=sec^2(x)-1

tan^{22} (x) = sec^{2} (x) - 1

tan^4(x)+tan^2(x)=tan(x)

tan^{4} (x) + tan^{2} (x) = tan (x)

sin^4(x)+sin^2(x)=sin^6(x)

sin^{4} (x) + sin^{2} (x) = sin^{6} (x)