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受欢迎的三角函数 >

tan^2(x)=sec(x)-1,0<= x<= 2pi

解答

tan^{2} (x) = sec (x) - 1, 0 \leq x \leq 2 π

解答

x = 0, x = 2 π

+1

度数

x = 0^{\circ}, x = 36 0^{\circ}

求解步骤

tan^{2} (x) = sec (x) - 1, 0 \leq x \leq 2 π

两边减去

sec (x) - 1

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

用替代法求解

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

令

: sec (x) = u

- u + u^{2} = 0

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

- u + u^{2} = 0

改写成标准形式

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

u^{2} - u = 0

使用求根公式求解

u^{2} - u = 0

二次方程求根公式:

若

a = 1, b = - 1, c = 0

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 1)^{2} - 4 \cdot 1 \cdot 0 = 1

(- 1)^{2} - 4 \cdot 1 \cdot 0

(- 1)^{2} = 1

(- 1)^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

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

= 1^{2}

使用法则

1^{a} = 1

= 1

4 \cdot 1 \cdot 0 = 0

4 \cdot 1 \cdot 0

使用法则

0 \cdot a = 0

= 0

= 1 - 0

数字相减:

1 - 0 = 1

= 1

使用法则

1 = 1

= 1

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

将解分隔开

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

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

\frac{- ( - 1 ) + 1}{2 \cdot 1}

使用法则

- (- a) = a

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

数字相加:

1 + 1 = 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{- ( - 1 ) - 1}{2 \cdot 1} : 0

\frac{- ( - 1 ) - 1}{2 \cdot 1}

使用法则

- (- a) = a

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

数字相减:

1 - 1 = 0

= \frac{0}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{0}{2}

使用法则

\frac{0}{a} = 0, a \neq = 0

= 0

二次方程组的解是:

u = 1, u = 0

u = sec (x)

代回

sec (x) = 1, sec (x) = 0

sec (x) = 1, sec (x) = 0

sec (x) = 1, 0 \leq x \leq 2 π : x = 0, x = 2 π

sec (x) = 1, 0 \leq x \leq 2 π

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

在

0 \leq x \leq 2 π

范围内的解

x = 0, x = 2 π

sec (x) = 0, 0 \leq x \leq 2 π :

无解

sec (x) = 0, 0 \leq x \leq 2 π

sec (x) \leq - 1 or sec (x) \geq 1

无解

合并所有解

x = 0, x = 2 π

作图

流行的例子

sin^2(θ)=2sin^2(θ/2)

sin^{2} (θ) = 2 sin^{2} (\frac{θ}{2})

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

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

0.26=sin(0.25pit)

0.26 = sin (0.25 π t)

4csc(θ+pi/3)=2

4 csc (θ + \frac{π}{3}) = 2

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

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