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受欢迎的三角函数 >

2tan^2(x)+1=sec^2(x)+sin^2(x)+cos^2(x)

解答

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

解答

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

+1

度数

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

求解步骤

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

两边减去

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

2 tan^{2} (x) + 1 - sec^{2} (x) - sin^{2} (x) - cos^{2} (x) = 0

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

= 1 - sec^{2} (x) + 2 tan^{2} (x) - 1

化简

1 - sec^{2} (x) + 2 tan^{2} (x) - 1 : 2 tan^{2} (x) - sec^{2} (x)

1 - sec^{2} (x) + 2 tan^{2} (x) - 1

对同类项分组

= - sec^{2} (x) + 2 tan^{2} (x) + 1 - 1

1 - 1 = 0

= 2 tan^{2} (x) - sec^{2} (x)

= 2 tan^{2} (x) - sec^{2} (x)

- sec^{2} (x) + 2 tan^{2} (x) = 0

分解

- sec^{2} (x) + 2 tan^{2} (x) : (2 tan (x) + sec (x)) (2 tan (x) - sec (x))

- sec^{2} (x) + 2 tan^{2} (x)

将

2 tan^{2} (x) - sec^{2} (x)

改写为

(2 tan (x))^{2} - sec^{2} (x)

2 tan^{2} (x) - sec^{2} (x)

使用根式运算法则:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} tan^{2} (x) - sec^{2} (x)

使用指数法则:

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

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

= (2 tan (x))^{2} - sec^{2} (x)

= (2 tan (x))^{2} - sec^{2} (x)

使用平方差公式:

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

(2 tan (x))^{2} - sec^{2} (x) = (2 tan (x) + sec (x)) (2 tan (x) - sec (x))

= (2 tan (x) + sec (x)) (2 tan (x) - sec (x))

(2 tan (x) + sec (x)) (2 tan (x) - sec (x)) = 0

分别求解每个部分

2 tan (x) + sec (x) = 0 or 2 tan (x) - sec (x) = 0

2 tan (x) + sec (x) = 0 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

2 tan (x) + sec (x) = 0

用 sin, cos 表示

sec (x) + 2 tan (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

乘

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

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

分式相乘:

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

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

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

使用法则

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

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

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

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

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

1 + sin (x) 2 = 0

将

1

到右边

1 + sin (x) 2 = 0

两边减去

1

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

化简

sin (x) 2 = - 1

sin (x) 2 = - 1

两边除以

2

sin (x) 2 = - 1

两边除以

2

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

化简

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

化简

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

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

约分:

2

= sin (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}

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

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

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

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

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

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

2 tan (x) - sec (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

2 tan (x) - sec (x) = 0

用 sin, cos 表示

- sec (x) + 2 tan (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

乘

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

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

分式相乘:

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

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

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

使用法则

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

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

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

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

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

- 1 + sin (x) 2 = 0

将

1

到右边

- 1 + sin (x) 2 = 0

两边加上

1

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

化简

sin (x) 2 = 1

sin (x) 2 = 1

两边除以

2

sin (x) 2 = 1

两边除以

2

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

化简

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

化简

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

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

约分:

2

= sin (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}

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

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

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

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

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

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

合并所有解

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

作图

流行的例子

cos^2(x)+5sin^2(x)=1

-1=-4sec(y)-sec^2(y)-5

2sin^2(x)+cos(2x)=4cos^2(x)