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受欢迎的三角函数 >

sin(x)cos(x)= 1/2 tan(x)

解答

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

解答

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

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

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

化简

sin (x) cos (x) - \frac{1}{2} tan (x) : \frac{2 sin ( x ) cos ( x ) - tan ( x )}{2}

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

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

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

分式相乘:

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

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

乘以:

1 \cdot tan (x) = tan (x)

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

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

将项转换为分式:

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

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

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 ( x ) cos ( x ) \cdot 2 - tan ( x )}{2}

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

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

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

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

化简

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

- \frac{sin ( x )}{cos ( x )} + 2 cos (x) sin (x)

将项转换为分式:

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

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

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 ( x ) + 2 cos ( x ) sin ( x ) cos ( x )}{cos ( x )}

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

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

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

2 cos (x) sin (x) cos (x)

使用指数法则:

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

cos (x) cos (x) = cos^{1 + 1} (x)

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

数字相加:

1 + 1 = 2

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

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

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

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

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

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

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

分解

- sin (x) + 2 cos^{2} (x) sin (x) : sin (x) (2 cos (x) + 1) (2 cos (x) - 1)

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

因式分解出通项

sin (x)

= sin (x) (- 1 + 2 cos^{2} (x))

分解

2 cos^{2} (x) - 1 : (2 cos (x) + 1) (2 cos (x) - 1)

2 cos^{2} (x) - 1

将

2 cos^{2} (x) - 1

改写为

(2 cos (x))^{2} - 1^{2}

2 cos^{2} (x) - 1

使用根式运算法则:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} cos^{2} (x) - 1

将

1

改写为

1^{2}

= (2)^{2} cos^{2} (x) - 1^{2}

使用指数法则:

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

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

= (2 cos (x))^{2} - 1^{2}

= (2 cos (x))^{2} - 1^{2}

使用平方差公式:

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

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

= (2 cos (x) + 1) (2 cos (x) - 1)

= sin (x) (2 cos (x) + 1) (2 cos (x) - 1)

sin (x) (2 cos (x) + 1) (2 cos (x) - 1) = 0

分别求解每个部分

sin (x) = 0 or 2 cos (x) + 1 = 0 or 2 cos (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

2 cos (x) + 1 = 0 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

2 cos (x) + 1 = 0

将

1

到右边

2 cos (x) + 1 = 0

两边减去

1

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

化简

2 cos (x) = - 1

2 cos (x) = - 1

两边除以

2

2 cos (x) = - 1

两边除以

2

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

化简

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

化简

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

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

约分:

2

= cos (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}

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

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

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

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

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}

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

2 cos (x) - 1 = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

2 cos (x) - 1 = 0

将

1

到右边

2 cos (x) - 1 = 0

两边加上

1

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

化简

2 cos (x) = 1

2 cos (x) = 1

两边除以

2

2 cos (x) = 1

两边除以

2

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

化简

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

化简

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

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

约分:

2

= cos (x)

化简

\frac{1}{2} : \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}

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

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

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

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

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}

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

作图

流行的例子

tan (x) = \frac{80}{50}

-cos(x)+4cos(2x)=0

- cos (x) + 4 cos (2 x) = 0

sin^2(x)+sin(x)-2=0,0<= x<2pi

sin^{2} (x) + sin (x) - 2 = 0, 0 \leq x < 2 π

sin^2(x)+5sin(x)=0

sin^{2} (x) + 5 sin (x) = 0

tan(α)=-4/3 , pi/2 <α<pi,cos(β)= 1/2 ,0<β< pi/2

tan (α) = - \frac{4}{3}, \frac{π}{2} < α < π, cos (β) = \frac{1}{2}, 0 < β < \frac{π}{2}