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受欢迎的三角函数 >

2tan(x)=tan(x)tan(x)

解答

2 tan (x) = tan (x) tan (x)

解答

x = πn, x = 1.10714 \dots + πn

+1

度数

x = 0^{\circ} + 18 0^{\circ} n, x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

求解步骤

2 tan (x) = tan (x) tan (x)

用替代法求解

2 tan (x) = tan (x) tan (x)

令

: tan (x) = u

2 u = uu

2 u = uu : u = 0, u = 2

2 u = uu

将

uu

para o lado esquerdo

2 u = uu

两边减去

uu

2 u - uu = uu - uu

化简

2 u - uu = 0

2 u - uu = 0

因式分解

2 u - uu : u (2 - u)

2 u - uu

因式分解出通项

u

= u (2 - u)

u (2 - u) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u = 0 or 2 - u = 0

解

2 - u = 0 : u = 2

2 - u = 0

将

2

到右边

2 - u = 0

两边减去

2

2 - u - 2 = 0 - 2

化简

- u = - 2

- u = - 2

两边除以

- 1

- u = - 2

两边除以

- 1

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

化简

u = 2

u = 2

二次方程组的解是:

u = 0, u = 2

u = tan (x)

代回

tan (x) = 0, tan (x) = 2

tan (x) = 0, tan (x) = 2

tan (x) = 0 : x = πn

tan (x) = 0

tan (x) = 0

的通解

tan (x)

周期表（周期为

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

解

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

tan (x) = 2 : x = arctan (2) + πn

tan (x) = 2

使用反三角函数性质

tan (x) = 2

tan (x) = 2

的通解

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

x = arctan (2) + πn

x = arctan (2) + πn

合并所有解

x = πn, x = arctan (2) + πn

以小数形式表示解

x = πn, x = 1.10714 \dots + πn

作图

流行的例子

(2tan(x))/(1-tan^2(x))=sqrt(3)

\frac{2 tan ( x )}{1 - tan ^{2} ( x )} = 3

tan (x) = \frac{59}{56}

cos(θ)= 5/(5sqrt(2))

cos (θ) = \frac{5}{5 2}

arccos(x)=arcsin(9/41)

arccos (x) = arcsin (\frac{9}{41})

solvefor x,-ysin(x)=0

so l v e f or x, - y sin (x) = 0