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受欢迎的三角函数 >

4sin^2(x)-6cos^2(x)=0

解答

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

解答

x = - 0.88607 \dots + πn, x = 0.88607 \dots + πn

+1

度数

x = - 50.76847 \dots^{\circ} + 18 0^{\circ} n, x = 50.76847 \dots^{\circ} + 18 0^{\circ} n

求解步骤

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

分解

4 sin^{2} (x) - 6 cos^{2} (x) : 2 (2 sin (x) + 3 cos (x)) (2 sin (x) - 3 cos (x))

4 sin^{2} (x) - 6 cos^{2} (x)

将

- 6

改写为

3 \cdot 2

将

4

改写为

2 \cdot 2

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

因式分解出通项

2

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

分解

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

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

将

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

改写为

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

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

使用根式运算法则:

a = (a)^{2}

2 = (2)^{2}

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

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

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

使用指数法则:

a^{m} b^{m} = (ab)^{m}

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

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

使用指数法则:

a^{m} b^{m} = (ab)^{m}

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

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

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

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

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

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

2 (2 sin (x) + 3 cos (x)) (2 sin (x) - 3 cos (x)) = 0

分别求解每个部分

2 sin (x) + 3 cos (x) = 0 or 2 sin (x) - 3 cos (x) = 0

2 sin (x) + 3 cos (x) = 0 : x = arctan (- \frac{3}{2}) + πn

2 sin (x) + 3 cos (x) = 0

使用三角恒等式改写

2 sin (x) + 3 cos (x) = 0

在两边除以

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

\frac{2 sin ( x ) + 3 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{2 sin ( x )}{cos ( x )} + 3 = 0

使用基本三角恒等式:

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

2 tan (x) + 3 = 0

2 tan (x) + 3 = 0

将

3

到右边

2 tan (x) + 3 = 0

两边减去

3

2 tan (x) + 3 - 3 = 0 - 3

化简

2 tan (x) = - 3

2 tan (x) = - 3

两边除以

2

2 tan (x) = - 3

两边除以

2

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

化简

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

化简

\frac{2 tan ( x )}{2} : tan (x)

\frac{2 tan ( x )}{2}

约分:

2

= tan (x)

化简

\frac{- 3}{2} : - \frac{3}{2}

\frac{- 3}{2}

使用分式法则:

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

= - \frac{3}{2}

合并相同指数项 :

\frac{x}{y} = \frac{x}{y}

= - \frac{3}{2}

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

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

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

使用反三角函数性质

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

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

的通解

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

x = arctan (- \frac{3}{2}) + πn

x = arctan (- \frac{3}{2}) + πn

2 sin (x) - 3 cos (x) = 0 : x = arctan (\frac{3}{2}) + πn

2 sin (x) - 3 cos (x) = 0

使用三角恒等式改写

2 sin (x) - 3 cos (x) = 0

在两边除以

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

\frac{2 sin ( x ) - 3 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{2 sin ( x )}{cos ( x )} - 3 = 0

使用基本三角恒等式:

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

2 tan (x) - 3 = 0

2 tan (x) - 3 = 0

将

3

到右边

2 tan (x) - 3 = 0

两边加上

3

2 tan (x) - 3 + 3 = 0 + 3

化简

2 tan (x) = 3

2 tan (x) = 3

两边除以

2

2 tan (x) = 3

两边除以

2

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

化简

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

化简

\frac{2 tan ( x )}{2} : tan (x)

\frac{2 tan ( x )}{2}

约分:

2

= tan (x)

化简

\frac{3}{2} : \frac{3}{2}

\frac{3}{2}

合并相同指数项 :

\frac{x}{y} = \frac{x}{y}

= \frac{3}{2}

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

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

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

使用反三角函数性质

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

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

的通解

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

x = arctan (\frac{3}{2}) + πn

x = arctan (\frac{3}{2}) + πn

合并所有解

x = arctan (- \frac{3}{2}) + πn, x = arctan (\frac{3}{2}) + πn

以小数形式表示解

x = - 0.88607 \dots + πn, x = 0.88607 \dots + πn

作图

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