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受欢迎的三角函数 >

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

解答

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

解答

x = - 0.61547 \dots + πn, x = 0.61547 \dots + πn

+1

度数

x = - 35.26438 \dots^{\circ} + 18 0^{\circ} n, x = 35.26438 \dots^{\circ} + 18 0^{\circ} n

求解步骤

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

两边减去

sin^{2} (x)

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

使用三角恒等式改写

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

使用倍角公式:

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

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

化简

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

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

分解

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

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

将

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

改写为

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

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

使用根式运算法则:

a = (a)^{2}

2 = (2)^{2}

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

使用指数法则:

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

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

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

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

使用平方差公式:

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

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

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

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

分别求解每个部分

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

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

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

使用三角恒等式改写

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

在两边除以

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

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

化简

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

使用基本三角恒等式:

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

1 + 2 tan (x) = 0

1 + 2 tan (x) = 0

将

1

到右边

1 + 2 tan (x) = 0

两边减去

1

1 + 2 tan (x) - 1 = 0 - 1

化简

2 tan (x) = - 1

2 tan (x) = - 1

两边除以

2

2 tan (x) = - 1

两边除以

2

\frac{2 tan ( x )}{2} = \frac{- 1}{2}

化简

\frac{2 tan ( x )}{2} = \frac{- 1}{2}

化简

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

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

约分:

2

= tan (x)

化简

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

使用分式法则:

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

= - \frac{1}{2}

- \frac{1}{2}

有理化:

- \frac{2}{2}

- \frac{1}{2}

乘以共轭根式

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

使用根式运算法则:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

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

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

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

使用反三角函数性质

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

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

的通解

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

x = arctan (- \frac{2}{2}) + πn

x = arctan (- \frac{2}{2}) + πn

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

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

使用三角恒等式改写

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

在两边除以

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

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

化简

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

使用基本三角恒等式:

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

1 - 2 tan (x) = 0

1 - 2 tan (x) = 0

将

1

到右边

1 - 2 tan (x) = 0

两边减去

1

1 - 2 tan (x) - 1 = 0 - 1

化简

- 2 tan (x) = - 1

- 2 tan (x) = - 1

两边除以

- 2

- 2 tan (x) = - 1

两边除以

- 2

\frac{- 2 tan ( x )}{- 2} = \frac{- 1}{- 2}

化简

\frac{- 2 tan ( x )}{- 2} = \frac{- 1}{- 2}

化简

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

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

使用分式法则:

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

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

约分:

2

= tan (x)

化简

\frac{- 1}{- 2} : \frac{2}{2}

\frac{- 1}{- 2}

使用分式法则:

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

= \frac{1}{2}

\frac{1}{2}

有理化:

\frac{2}{2}

\frac{1}{2}

乘以共轭根式

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

使用根式运算法则:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

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

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

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

使用反三角函数性质

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

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

的通解

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

x = arctan (\frac{2}{2}) + πn

x = arctan (\frac{2}{2}) + πn

合并所有解

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

以小数形式表示解

x = - 0.61547 \dots + πn, x = 0.61547 \dots + πn

作图

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