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受欢迎的三角函数 >

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

解答

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

解答

a = 2 πn, a = π + 2 πn

+1

度数

a = 0^{\circ} + 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

tan (a) sin (a)

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

使用三角恒等式改写

(1 + cos (a)) (1 - cos (a)) - sin (a) tan (a)

1 - cos (a) = \frac{sin ^{2} ( a )}{1 + cos ( a )}

1 - cos (a)

乘以

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

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

使用平方差公式:

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

(1 - cos (a)) (1 + cos (a)) = 1 - cos^{2} (a)

= \frac{1 - cos ^{2} ( a )}{1 + cos ( a )}

使用毕达哥拉斯恒等式:

1 = cos^{2} (a) + sin^{2} (a)

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

= \frac{sin ^{2} ( a )}{1 + cos ( a )}

= (1 + cos (a)) \frac{sin ^{2} ( a )}{1 + cos ( a )} - sin (a) tan (a)

(1 + cos (a)) \frac{sin ^{2} ( a )}{1 + cos ( a )} = sin^{2} (a)

(1 + cos (a)) \frac{sin ^{2} ( a )}{1 + cos ( a )}

分式相乘:

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

= \frac{sin ^{2} ( a ) ( 1 + cos ( a ) )}{1 + cos ( a )}

约分:

1 + cos (a)

= sin^{2} (a)

= sin^{2} (a) - sin (a) tan (a)

sin^{2} (a) - sin (a) tan (a) = 0

分解

sin^{2} (a) - sin (a) tan (a) : sin (a) (sin (a) - tan (a))

sin^{2} (a) - sin (a) tan (a)

使用指数法则:

a^{b + c} = a^{b} a^{c}

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

= sin (a) sin (a) - sin (a) tan (a)

因式分解出通项

sin (a)

= sin (a) (sin (a) - tan (a))

sin (a) (sin (a) - tan (a)) = 0

分别求解每个部分

sin (a) = 0 or sin (a) - tan (a) = 0

sin (a) = 0 : a = 2 πn, a = π + 2 πn

sin (a) = 0

sin (a) = 0

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 = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn, a = π + 2 πn

解

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn, a = π + 2 πn

sin (a) - tan (a) = 0 : a = 2 πn, a = π + 2 πn

sin (a) - tan (a) = 0

用 sin, cos 表示

sin (a) - tan (a)

使用基本三角恒等式:

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

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

化简

sin (a) - \frac{sin ( a )}{cos ( a )} : \frac{sin ( a ) cos ( a ) - sin ( a )}{cos ( a )}

sin (a) - \frac{sin ( a )}{cos ( a )}

将项转换为分式:

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

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

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{sin ( a ) cos ( a ) - sin ( a )}{cos ( a )}

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

\frac{- sin ( a ) + cos ( a ) sin ( a )}{cos ( a )} = 0

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

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

分解

- sin (a) + cos (a) sin (a) : sin (a) (cos (a) - 1)

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

因式分解出通项

sin (a)

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

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

分别求解每个部分

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

sin (a) = 0 : a = 2 πn, a = π + 2 πn

sin (a) = 0

sin (a) = 0

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 = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn, a = π + 2 πn

解

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn, a = π + 2 πn

cos (a) - 1 = 0 : a = 2 πn

cos (a) - 1 = 0

将

1

到右边

cos (a) - 1 = 0

两边加上

1

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

化简

cos (a) = 1

cos (a) = 1

cos (a) = 1

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

a = 0 + 2 πn

a = 0 + 2 πn

解

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn

合并所有解

a = 2 πn, a = π + 2 πn

合并所有解

a = 2 πn, a = π + 2 πn

作图

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