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受欢迎的三角函数 >

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

解答

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

解答

x = - 0.46364 \dots + πn, x = 0.46364 \dots + πn

+1

度数

x = - 26.56505 \dots^{\circ} + 18 0^{\circ} n, x = 26.56505 \dots^{\circ} + 18 0^{\circ} n

求解步骤

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

分解

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

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

将

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

改写为

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

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

将

4

改写为

2^{2}

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

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

使用平方差公式:

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

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

分别求解每个部分

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

2 sin (x) + cos (x) = 0 : x = arctan (- \frac{1}{2}) + πn

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

使用三角恒等式改写

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

在两边除以

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

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

化简

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

使用基本三角恒等式:

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

2 tan (x) + 1 = 0

2 tan (x) + 1 = 0

将

1

到右边

2 tan (x) + 1 = 0

两边减去

1

2 tan (x) + 1 - 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

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

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

使用反三角函数性质

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

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

的通解

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

x = arctan (- \frac{1}{2}) + πn

x = arctan (- \frac{1}{2}) + πn

2 sin (x) - cos (x) = 0 : x = arctan (\frac{1}{2}) + πn

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

使用三角恒等式改写

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

在两边除以

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

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

化简

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

使用基本三角恒等式:

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

2 tan (x) - 1 = 0

2 tan (x) - 1 = 0

将

1

到右边

2 tan (x) - 1 = 0

两边加上

1

2 tan (x) - 1 + 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

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

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

使用反三角函数性质

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

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

的通解

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

x = arctan (\frac{1}{2}) + πn

x = arctan (\frac{1}{2}) + πn

合并所有解

x = arctan (- \frac{1}{2}) + πn, x = arctan (\frac{1}{2}) + πn

以小数形式表示解

x = - 0.46364 \dots + πn, x = 0.46364 \dots + πn

作图

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