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受欢迎的三角函数 >

1/2 tan^2(x)=1-sec(x)

解答

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

解答

x = 2 πn, x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 109.47122 \dots^{\circ} + 36 0^{\circ} n, x = - 109.47122 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

1 - sec (x)

\frac{1}{2} tan^{2} (x) - 1 + sec (x) = 0

化简

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

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

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

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

分式相乘:

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

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

乘以:

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

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

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

将项转换为分式:

1 = \frac{1 \cdot 2}{2}, sec (x) = \frac{sec ( x ) 2}{2}

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

数字相乘:

1 \cdot 2 = 2

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

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

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

化简

- 2 + sec^{2} (x) - 1 + 2 sec (x) : sec^{2} (x) + 2 sec (x) - 3

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

对同类项分组

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

数字相减:

- 2 - 1 = - 3

= sec^{2} (x) + 2 sec (x) - 3

= sec^{2} (x) + 2 sec (x) - 3

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

用替代法求解

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

令

: sec (x) = u

- 3 + u^{2} + 2 u = 0

- 3 + u^{2} + 2 u = 0 : u = 1, u = - 3

- 3 + u^{2} + 2 u = 0

改写成标准形式

a x^{2} + b x + c = 0

u^{2} + 2 u - 3 = 0

使用求根公式求解

u^{2} + 2 u - 3 = 0

二次方程求根公式:

若

a = 1, b = 2, c = - 3

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 1 \cdot ( - 3 )}{2 \cdot 1}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 1 \cdot ( - 3 )}{2 \cdot 1}

2^{2} - 4 \cdot 1 \cdot (- 3) = 4

2^{2} - 4 \cdot 1 \cdot (- 3)

使用法则

- (- a) = a

= 2^{2} + 4 \cdot 1 \cdot 3

数字相乘:

4 \cdot 1 \cdot 3 = 12

= 2^{2} + 12

2^{2} = 4

= 4 + 12

数字相加:

4 + 12 = 16

= 16

因式分解数字:

16 = 4^{2}

= 4^{2}

使用根式运算法则:

n a^{n} = a

4^{2} = 4

= 4

u_{1, 2} = \frac{- 2 \pm 4}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- 2 + 4}{2 \cdot 1}, u_{2} = \frac{- 2 - 4}{2 \cdot 1}

u = \frac{- 2 + 4}{2 \cdot 1} : 1

\frac{- 2 + 4}{2 \cdot 1}

数字相加/相减:

- 2 + 4 = 2

= \frac{2}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{2}{2}

使用法则

\frac{a}{a} = 1

= 1

u = \frac{- 2 - 4}{2 \cdot 1} : - 3

\frac{- 2 - 4}{2 \cdot 1}

数字相减:

- 2 - 4 = - 6

= \frac{- 6}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 6}{2}

使用分式法则:

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

= - \frac{6}{2}

数字相除:

\frac{6}{2} = 3

= - 3

二次方程组的解是:

u = 1, u = - 3

u = sec (x)

代回

sec (x) = 1, sec (x) = - 3

sec (x) = 1, sec (x) = - 3

sec (x) = 1 : x = 2 πn

sec (x) = 1

sec (x) = 1

的通解

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 2 πn

x = 0 + 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

sec (x) = - 3 : x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

sec (x) = - 3

使用反三角函数性质

sec (x) = - 3

sec (x) = - 3

的通解

sec (x) = - a \Rightarrow x = arcsec (- a) + 2 πn, x = - arcsec (- a) + 2 πn

x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

合并所有解

x = 2 πn, x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

以小数形式表示解

x = 2 πn, x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

作图

流行的例子

sin(2x)=sin^2(x)

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

2sec^2(x)+3(1/(cos(x)))=2

2 sec^{2} (x) + 3 (\frac{1}{cos ( x )}) = 2

cot^{2} (θ) - 3 = 0

tan(θ)=2,sin(θ)

tan (θ) = 2, sin (θ)

7sin^2(x)+14sin(x)+7=0

7 sin^{2} (x) + 14 sin (x) + 7 = 0