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受欢迎的三角函数 >

(sec^2(a))cos^2(a)=sin^2(a)

解答

(sec^{2} (a)) cos^{2} (a) = sin^{2} (a)

解答

a \in R 无解

求解步骤

(sec^{2} (a)) cos^{2} (a) = sin^{2} (a)

两边减去

sin^{2} (a)

sec^{2} (a) cos^{2} (a) - sin^{2} (a) = 0

分解

sec^{2} (a) cos^{2} (a) - sin^{2} (a) : (sec (a) cos (a) + sin (a)) (sec (a) cos (a) - sin (a))

sec^{2} (a) cos^{2} (a) - sin^{2} (a)

将

sec^{2} (a) cos^{2} (a)

改写为

(sec (a) cos (a))^{2}

sec^{2} (a) cos^{2} (a)

使用指数法则:

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

sec^{2} (a) cos^{2} (a) = (sec (a) cos (a))^{2}

= (sec (a) cos (a))^{2}

= (sec (a) cos (a))^{2} - sin^{2} (a)

使用平方差公式:

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

(sec (a) cos (a))^{2} - sin^{2} (a) = (sec (a) cos (a) + sin (a)) (sec (a) cos (a) - sin (a))

= (sec (a) cos (a) + sin (a)) (sec (a) cos (a) - sin (a))

(sec (a) cos (a) + sin (a)) (sec (a) cos (a) - sin (a)) = 0

使用三角恒等式改写

(sec (a) cos (a) + sin (a)) (sec (a) cos (a) - sin (a))

sec (a) cos (a) = 1

sec (a) cos (a)

用 sin, cos 表示

sec (a) cos (a)

使用基本三角恒等式:

sec (a) = \frac{1}{cos ( a )}

= \frac{1}{cos ( a )} cos (a)

\frac{1}{cos ( a )} cos (a) = 1

\frac{1}{cos ( a )} cos (a)

分式相乘:

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

= \frac{1 cos ( a )}{cos ( a )}

约分:

cos (a)

= 1

= 1

= (- sin (a) + 1) (sin (a) + 1)

(- sin (a) + 1) (sin (a) + 1) = 0

分别求解每个部分

- sin (a) + 1 = 0 or sin (a) + 1 = 0

- sin (a) + 1 = 0 : a = \frac{π}{2} + 2 πn

- sin (a) + 1 = 0

将

1

到右边

- sin (a) + 1 = 0

两边减去

1

- sin (a) + 1 - 1 = 0 - 1

化简

- sin (a) = - 1

- sin (a) = - 1

两边除以

- 1

- sin (a) = - 1

两边除以

- 1

\frac{- sin ( a )}{- 1} = \frac{- 1}{- 1}

化简

sin (a) = 1

sin (a) = 1

sin (a) = 1

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

a = \frac{π}{2} + 2 πn

a = \frac{π}{2} + 2 πn

sin (a) + 1 = 0 : a = \frac{3 π}{2} + 2 πn

sin (a) + 1 = 0

将

1

到右边

sin (a) + 1 = 0

两边减去

1

sin (a) + 1 - 1 = 0 - 1

化简

sin (a) = - 1

sin (a) = - 1

sin (a) = - 1

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

a = \frac{3 π}{2} + 2 πn

a = \frac{3 π}{2} + 2 πn

合并所有解

a = \frac{π}{2} + 2 πn, a = \frac{3 π}{2} + 2 πn

因为方程对以下值无定义:

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn

a \in R 无解

作图

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