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受欢迎的三角函数 >

tan(x+pi/4)-tan(x-pi/4)=3

解答

tan (x + \frac{π}{4}) - tan (x - \frac{π}{4}) = 3

解答

x = - 0.42053 \dots + πn, x = 0.42053 \dots + πn

+1

度数

x = - 24.09484 \dots^{\circ} + 18 0^{\circ} n, x = 24.09484 \dots^{\circ} + 18 0^{\circ} n

求解步骤

tan (x + \frac{π}{4}) - tan (x - \frac{π}{4}) = 3

使用三角恒等式改写

tan (x + \frac{π}{4}) - tan (x - \frac{π}{4}) = 3

使用三角恒等式改写

tan (x - \frac{π}{4})

使用基本三角恒等式:

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

= \frac{sin ( x - \frac{π}{4} )}{cos ( x - \frac{π}{4} )}

使用角差恒等式:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= \frac{sin ( x ) cos ( \frac{π}{4} ) - cos ( x ) sin ( \frac{π}{4} )}{cos ( x - \frac{π}{4} )}

使用角差恒等式:

cos (s - t) = cos (s) cos (t) + sin (s) sin (t)

= \frac{sin ( x ) cos ( \frac{π}{4} ) - cos ( x ) sin ( \frac{π}{4} )}{cos ( x ) cos ( \frac{π}{4} ) + sin ( x ) sin ( \frac{π}{4} )}

化简

\frac{sin ( x ) cos ( \frac{π}{4} ) - cos ( x ) sin ( \frac{π}{4} )}{cos ( x ) cos ( \frac{π}{4} ) + sin ( x ) sin ( \frac{π}{4} )} : \frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )}

\frac{sin ( x ) cos ( \frac{π}{4} ) - cos ( x ) sin ( \frac{π}{4} )}{cos ( x ) cos ( \frac{π}{4} ) + sin ( x ) sin ( \frac{π}{4} )}

sin (x) cos (\frac{π}{4}) - cos (x) sin (\frac{π}{4}) = \frac{2}{2} sin (x) - \frac{2}{2} cos (x)

sin (x) cos (\frac{π}{4}) - cos (x) sin (\frac{π}{4})

化简

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

使用以下普通恒等式:

cos (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

= \frac{2}{2} sin (x) - sin (\frac{π}{4}) cos (x)

化简

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

使用以下普通恒等式:

sin (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

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

= \frac{\frac{2}{2} sin ( x ) - \frac{2}{2} cos ( x )}{cos ( \frac{π}{4} ) cos ( x ) + sin ( \frac{π}{4} ) sin ( x )}

cos (x) cos (\frac{π}{4}) + sin (x) sin (\frac{π}{4}) = \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

cos (x) cos (\frac{π}{4}) + sin (x) sin (\frac{π}{4})

化简

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

使用以下普通恒等式:

cos (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

= \frac{2}{2} cos (x) + sin (\frac{π}{4}) sin (x)

化简

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

使用以下普通恒等式:

sin (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

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

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

乘

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

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

分式相乘:

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

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

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

乘

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

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

分式相乘:

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

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

= \frac{\frac{2}{2} sin ( x ) - \frac{2}{2} cos ( x )}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

乘

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

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

分式相乘:

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

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

= \frac{\frac{2 s i n ( x )}{2} - \frac{2}{2} cos ( x )}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

乘

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

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

分式相乘:

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

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

= \frac{\frac{2 s i n ( x )}{2} - \frac{2 c o s ( x )}{2}}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

合并分式

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

使用法则

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

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

= \frac{\frac{2 s i n ( x )}{2} - \frac{2 c o s ( x )}{2}}{\frac{2 c o s ( x ) + 2 s i n ( x )}{2}}

合并分式

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

使用法则

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

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

= \frac{\frac{2 s i n ( x ) - 2 c o s ( x )}{2}}{\frac{2 c o s ( x ) + 2 s i n ( x )}{2}}

分式相除:

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

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

约分:

2

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

因式分解出通项

2

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

因式分解出通项

2

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

约分:

2

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

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

使用基本三角恒等式:

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

= \frac{sin ( x + \frac{π}{4} )}{cos ( x + \frac{π}{4} )}

使用角和恒等式:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= \frac{sin ( x ) cos ( \frac{π}{4} ) + cos ( x ) sin ( \frac{π}{4} )}{cos ( x + \frac{π}{4} )}

使用角和恒等式:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= \frac{sin ( x ) cos ( \frac{π}{4} ) + cos ( x ) sin ( \frac{π}{4} )}{cos ( x ) cos ( \frac{π}{4} ) - sin ( x ) sin ( \frac{π}{4} )}

化简

\frac{sin ( x ) cos ( \frac{π}{4} ) + cos ( x ) sin ( \frac{π}{4} )}{cos ( x ) cos ( \frac{π}{4} ) - sin ( x ) sin ( \frac{π}{4} )} : \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )}

\frac{sin ( x ) cos ( \frac{π}{4} ) + cos ( x ) sin ( \frac{π}{4} )}{cos ( x ) cos ( \frac{π}{4} ) - sin ( x ) sin ( \frac{π}{4} )}

sin (x) cos (\frac{π}{4}) + cos (x) sin (\frac{π}{4}) = \frac{2}{2} sin (x) + \frac{2}{2} cos (x)

sin (x) cos (\frac{π}{4}) + cos (x) sin (\frac{π}{4})

化简

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

使用以下普通恒等式:

cos (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

= \frac{2}{2} sin (x) + sin (\frac{π}{4}) cos (x)

化简

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

使用以下普通恒等式:

sin (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

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

= \frac{\frac{2}{2} sin ( x ) + \frac{2}{2} cos ( x )}{cos ( \frac{π}{4} ) cos ( x ) - sin ( \frac{π}{4} ) sin ( x )}

cos (x) cos (\frac{π}{4}) - sin (x) sin (\frac{π}{4}) = \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

cos (x) cos (\frac{π}{4}) - sin (x) sin (\frac{π}{4})

化简

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

使用以下普通恒等式:

cos (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

= \frac{2}{2} cos (x) - sin (\frac{π}{4}) sin (x)

化简

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

使用以下普通恒等式:

sin (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

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

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

乘

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

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

分式相乘:

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

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

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

乘

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

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

分式相乘:

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

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

= \frac{\frac{2}{2} sin ( x ) + \frac{2}{2} cos ( x )}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

乘

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

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

分式相乘:

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

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

= \frac{\frac{2 s i n ( x )}{2} + \frac{2}{2} cos ( x )}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

乘

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

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

分式相乘:

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

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

= \frac{\frac{2 s i n ( x )}{2} + \frac{2 c o s ( x )}{2}}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

合并分式

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

使用法则

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

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

= \frac{\frac{2 s i n ( x )}{2} + \frac{2 c o s ( x )}{2}}{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}

合并分式

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

使用法则

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

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

= \frac{\frac{2 s i n ( x ) + 2 c o s ( x )}{2}}{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}

分式相除:

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

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

约分:

2

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

因式分解出通项

2

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

因式分解出通项

2

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

约分:

2

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

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

\frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )} - \frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )} = 3

化简

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

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

cos (x) - sin (x), cos (x) + sin (x)

的最小公倍数:

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

cos (x) - sin (x), cos (x) + sin (x)

最小公倍数 (LCM)

计算出由出现在

cos (x) - sin (x)

或

cos (x) + sin (x)

中的因子组成的表达式

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

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

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

对于

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

将分母和分子乘以

cos (x) + sin (x)

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

对于

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

将分母和分子乘以

cos (x) - sin (x)

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

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

乘开

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

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

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

使用完全平方公式:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = sin (x), b = cos (x)

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

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

乘开

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

乘开

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

(sin (x) - cos (x)) (cos (x) - sin (x))

使用 FOIL 方法:

(a + b) (c + d) = a c + a d + b c + b d

a = sin (x), b = - cos (x), c = cos (x), d = - sin (x)

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

使用加减运算法则

+ (- a) = - a, (- a) (- b) = ab

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

化简

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

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

同类项相加:

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

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

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

sin (x) sin (x)

使用指数法则:

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

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

= sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sin^{2} (x)

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

cos (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)

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

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

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

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

打开括号

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

使用加减运算法则

- (- a) = a, - (a) = - a

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

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

化简

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

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

同类项相加:

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

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

同类项相加:

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

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

同类项相加:

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

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

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

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

\frac{2 sin ^{2} ( x ) + 2 cos ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} = 3

\frac{2 sin ^{2} ( x ) + 2 cos ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} = 3

两边减去

3

\frac{2 sin ^{2} ( x ) + 2 cos ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} - 3 = 0

化简

\frac{2 sin ^{2} ( x ) + 2 cos ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} - 3 : \frac{5 sin ^{2} ( x ) - cos ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

\frac{2 sin ^{2} ( x ) + 2 cos ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} - 3

将项转换为分式:

3 = \frac{3 ( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

= \frac{2 sin ^{2} ( x ) + 2 cos ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} - \frac{3 ( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( 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{2 sin ^{2} ( x ) + 2 cos ^{2} ( x ) - 3 ( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

乘开

2 sin^{2} (x) + 2 cos^{2} (x) - 3 (cos (x) - sin (x)) (cos (x) + sin (x)) : 5 sin^{2} (x) - cos^{2} (x)

2 sin^{2} (x) + 2 cos^{2} (x) - 3 (cos (x) - sin (x)) (cos (x) + sin (x))

乘开

- 3 (cos (x) - sin (x)) (cos (x) + sin (x)) : - 3 cos^{2} (x) + 3 sin^{2} (x)

乘开

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

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

使用平方差公式:

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

a = cos (x), b = sin (x)

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

= - 3 (cos^{2} (x) - sin^{2} (x))

乘开

- 3 (cos^{2} (x) - sin^{2} (x)) : - 3 cos^{2} (x) + 3 sin^{2} (x)

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

使用分配律:

a (b - c) = ab - a c

a = - 3, b = cos^{2} (x), c = sin^{2} (x)

= - 3 cos^{2} (x) - (- 3) sin^{2} (x)

使用加减运算法则

- (- a) = a

= - 3 cos^{2} (x) + 3 sin^{2} (x)

= - 3 cos^{2} (x) + 3 sin^{2} (x)

= 2 sin^{2} (x) + 2 cos^{2} (x) - 3 cos^{2} (x) + 3 sin^{2} (x)

化简

2 sin^{2} (x) + 2 cos^{2} (x) - 3 cos^{2} (x) + 3 sin^{2} (x) : 5 sin^{2} (x) - cos^{2} (x)

2 sin^{2} (x) + 2 cos^{2} (x) - 3 cos^{2} (x) + 3 sin^{2} (x)

同类项相加:

2 cos^{2} (x) - 3 cos^{2} (x) = - cos^{2} (x)

= 2 sin^{2} (x) - cos^{2} (x) + 3 sin^{2} (x)

同类项相加:

2 sin^{2} (x) + 3 sin^{2} (x) = 5 sin^{2} (x)

= 5 sin^{2} (x) - cos^{2} (x)

= 5 sin^{2} (x) - cos^{2} (x)

= \frac{5 sin ^{2} ( x ) - cos ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

\frac{5 sin ^{2} ( x ) - cos ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} = 0

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

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

分解

5 sin^{2} (x) - cos^{2} (x) : (5 sin (x) + cos (x)) (5 sin (x) - cos (x))

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

将

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

改写为

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

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

使用根式运算法则:

a = (a)^{2}

5 = (5)^{2}

= (5)^{2} sin^{2} (x) - cos^{2} (x)

使用指数法则:

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

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

= (5 sin (x))^{2} - cos^{2} (x)

= (5 sin (x))^{2} - cos^{2} (x)

使用平方差公式:

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

(5 sin (x))^{2} - cos^{2} (x) = (5 sin (x) + cos (x)) (5 sin (x) - cos (x))

= (5 sin (x) + cos (x)) (5 sin (x) - cos (x))

(5 sin (x) + cos (x)) (5 sin (x) - cos (x)) = 0

分别求解每个部分

5 sin (x) + cos (x) = 0 or 5 sin (x) - cos (x) = 0

5 sin (x) + cos (x) = 0 : x = arctan (- \frac{5}{5}) + πn

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

使用三角恒等式改写

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

在两边除以

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

\frac{5 sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{5 sin ( x )}{cos ( x )} + 1 = 0

使用基本三角恒等式:

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

5 tan (x) + 1 = 0

5 tan (x) + 1 = 0

将

1

到右边

5 tan (x) + 1 = 0

两边减去

1

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

化简

5 tan (x) = - 1

5 tan (x) = - 1

两边除以

5

5 tan (x) = - 1

两边除以

5

\frac{5 tan ( x )}{5} = \frac{- 1}{5}

化简

\frac{5 tan ( x )}{5} = \frac{- 1}{5}

化简

\frac{5 tan ( x )}{5} : tan (x)

\frac{5 tan ( x )}{5}

约分:

5

= tan (x)

化简

\frac{- 1}{5} : - \frac{5}{5}

\frac{- 1}{5}

使用分式法则:

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

= - \frac{1}{5}

- \frac{1}{5}

有理化:

- \frac{5}{5}

- \frac{1}{5}

乘以共轭根式

\frac{5}{5}

= - \frac{1 \cdot 5}{5 5}

1 \cdot 5 = 5

55 = 5

55

使用根式运算法则:

a a = a

55 = 5

= 5

= - \frac{5}{5}

= - \frac{5}{5}

tan (x) = - \frac{5}{5}

tan (x) = - \frac{5}{5}

tan (x) = - \frac{5}{5}

使用反三角函数性质

tan (x) = - \frac{5}{5}

tan (x) = - \frac{5}{5}

的通解

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

x = arctan (- \frac{5}{5}) + πn

x = arctan (- \frac{5}{5}) + πn

5 sin (x) - cos (x) = 0 : x = arctan (\frac{5}{5}) + πn

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

使用三角恒等式改写

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

在两边除以

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

\frac{5 sin ( x ) - cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

化简

\frac{5 sin ( x )}{cos ( x )} - 1 = 0

使用基本三角恒等式:

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

5 tan (x) - 1 = 0

5 tan (x) - 1 = 0

将

1

到右边

5 tan (x) - 1 = 0

两边加上

1

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

化简

5 tan (x) = 1

5 tan (x) = 1

两边除以

5

5 tan (x) = 1

两边除以

5

\frac{5 tan ( x )}{5} = \frac{1}{5}

化简

\frac{5 tan ( x )}{5} = \frac{1}{5}

化简

\frac{5 tan ( x )}{5} : tan (x)

\frac{5 tan ( x )}{5}

约分:

5

= tan (x)

化简

\frac{1}{5} : \frac{5}{5}

\frac{1}{5}

乘以共轭根式

\frac{5}{5}

= \frac{1 \cdot 5}{5 5}

1 \cdot 5 = 5

55 = 5

55

使用根式运算法则:

a a = a

55 = 5

= 5

= \frac{5}{5}

tan (x) = \frac{5}{5}

tan (x) = \frac{5}{5}

tan (x) = \frac{5}{5}

使用反三角函数性质

tan (x) = \frac{5}{5}

tan (x) = \frac{5}{5}

的通解

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

x = arctan (\frac{5}{5}) + πn

x = arctan (\frac{5}{5}) + πn

合并所有解

x = arctan (- \frac{5}{5}) + πn, x = arctan (\frac{5}{5}) + πn

以小数形式表示解

x = - 0.42053 \dots + πn, x = 0.42053 \dots + πn

作图

流行的例子

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2sin^2(θ)-3sin(θ)=-1

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cos (kπ) = 0

sin (θ) = - 0.8

cos(x/3-pi/4)= 1/2

cos (\frac{x}{3} - \frac{π}{4}) = \frac{1}{2}