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受欢迎的三角函数 >

(4sin(x)-3cos(x))/(3sin(x)+4cos(x))=3

解答

\frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} = 3

解答

x = - 1.24904 \dots + πn

+1

度数

x = - 71.56505 \dots^{\circ} + 18 0^{\circ} n

求解步骤

\frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} = 3

两边减去

3

\frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} - 3 = 0

化简

\frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} - 3 : \frac{- 5 sin ( x ) - 15 cos ( x )}{3 sin ( x ) + 4 cos ( x )}

\frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} - 3

将项转换为分式:

3 = \frac{3 ( 3 sin ( x ) + 4 cos ( x ) )}{3 sin ( x ) + 4 cos ( x )}

= \frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} - \frac{3 ( 3 sin ( x ) + 4 cos ( x ) )}{3 sin ( x ) + 4 cos ( x )}

因为分母相等，所以合并分式:

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

= \frac{4 sin ( x ) - 3 cos ( x ) - 3 ( 3 sin ( x ) + 4 cos ( x ) )}{3 sin ( x ) + 4 cos ( x )}

乘开

4 sin (x) - 3 cos (x) - 3 (3 sin (x) + 4 cos (x)) : - 5 sin (x) - 15 cos (x)

4 sin (x) - 3 cos (x) - 3 (3 sin (x) + 4 cos (x))

乘开

- 3 (3 sin (x) + 4 cos (x)) : - 9 sin (x) - 12 cos (x)

- 3 (3 sin (x) + 4 cos (x))

使用分配律:

a (b + c) = ab + a c

a = - 3, b = 3 sin (x), c = 4 cos (x)

= - 3 \cdot 3 sin (x) + (- 3) \cdot 4 cos (x)

使用加减运算法则

+ (- a) = - a

= - 3 \cdot 3 sin (x) - 3 \cdot 4 cos (x)

化简

- 3 \cdot 3 sin (x) - 3 \cdot 4 cos (x) : - 9 sin (x) - 12 cos (x)

- 3 \cdot 3 sin (x) - 3 \cdot 4 cos (x)

数字相乘:

3 \cdot 3 = 9

= - 9 sin (x) - 3 \cdot 4 cos (x)

数字相乘:

3 \cdot 4 = 12

= - 9 sin (x) - 12 cos (x)

= - 9 sin (x) - 12 cos (x)

= 4 sin (x) - 3 cos (x) - 9 sin (x) - 12 cos (x)

化简

4 sin (x) - 3 cos (x) - 9 sin (x) - 12 cos (x) : - 5 sin (x) - 15 cos (x)

4 sin (x) - 3 cos (x) - 9 sin (x) - 12 cos (x)

同类项相加:

- 3 cos (x) - 12 cos (x) = - 15 cos (x)

= 4 sin (x) - 15 cos (x) - 9 sin (x)

同类项相加:

4 sin (x) - 9 sin (x) = - 5 sin (x)

= - 5 sin (x) - 15 cos (x)

= - 5 sin (x) - 15 cos (x)

= \frac{- 5 sin ( x ) - 15 cos ( x )}{3 sin ( x ) + 4 cos ( x )}

\frac{- 5 sin ( x ) - 15 cos ( x )}{3 sin ( x ) + 4 cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 5 sin (x) - 15 cos (x) = 0

使用三角恒等式改写

- 5 sin (x) - 15 cos (x) = 0

在两边除以

cos (x), cos (x) \neq = 0

\frac{- 5 sin ( x ) - 15 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

- \frac{5 sin ( x )}{cos ( x )} - 15 = 0

使用基本三角恒等式:

\frac{sin ( x )}{cos ( x )} = tan (x)

- 5 tan (x) - 15 = 0

- 5 tan (x) - 15 = 0

将

15

到右边

- 5 tan (x) - 15 = 0

两边加上

15

- 5 tan (x) - 15 + 15 = 0 + 15

化简

- 5 tan (x) = 15

- 5 tan (x) = 15

两边除以

- 5

- 5 tan (x) = 15

两边除以

- 5

\frac{- 5 tan ( x )}{- 5} = \frac{15}{- 5}

化简

\frac{- 5 tan ( x )}{- 5} = \frac{15}{- 5}

化简

\frac{- 5 tan ( x )}{- 5} : tan (x)

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

使用分式法则:

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

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

数字相除:

\frac{5}{5} = 1

= tan (x)

化简

\frac{15}{- 5} : - 3

\frac{15}{- 5}

使用分式法则:

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

= - \frac{15}{5}

数字相除:

\frac{15}{5} = 3

= - 3

tan (x) = - 3

tan (x) = - 3

tan (x) = - 3

使用反三角函数性质

tan (x) = - 3

tan (x) = - 3

的通解

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- 3) + πn

x = arctan (- 3) + πn

以小数形式表示解

x = - 1.24904 \dots + πn

作图

流行的例子

7=10sin(40(0-h))+17

7 = 10 sin (40 (0 - h)) + 17

sec (x) = - \frac{5}{3}

2sin(2x)+2cos(x)=0

2 sin (2 x) + 2 cos (x) = 0

cosh (x) = \frac{3}{2}

(1+tan(t))/(sin(t))=0

\frac{1 + tan ( t )}{sin ( t )} = 0