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受欢迎的三角函数 >

4+2tan^2(x)=5tan^2(x)+3

解答

4 + 2 tan^{2} (x) = 5 tan^{2} (x) + 3

解答

x = 0.52359 \dots + πn, x = - 0.52359 \dots + πn

+1

度数

x = 3 0^{\circ} + 18 0^{\circ} n, x = - 3 0^{\circ} + 18 0^{\circ} n

求解步骤

4 + 2 tan^{2} (x) = 5 tan^{2} (x) + 3

用替代法求解

4 + 2 tan^{2} (x) = 5 tan^{2} (x) + 3

令

: tan (x) = u

4 + 2 u^{2} = 5 u^{2} + 3

4 + 2 u^{2} = 5 u^{2} + 3 : u = \frac{1}{3}, u = - \frac{1}{3}

4 + 2 u^{2} = 5 u^{2} + 3

将

4

到右边

4 + 2 u^{2} = 5 u^{2} + 3

两边减去

4

4 + 2 u^{2} - 4 = 5 u^{2} + 3 - 4

化简

2 u^{2} = 5 u^{2} - 1

2 u^{2} = 5 u^{2} - 1

将

5 u^{2}

para o lado esquerdo

2 u^{2} = 5 u^{2} - 1

两边减去

5 u^{2}

2 u^{2} - 5 u^{2} = 5 u^{2} - 1 - 5 u^{2}

化简

- 3 u^{2} = - 1

- 3 u^{2} = - 1

两边除以

- 3

- 3 u^{2} = - 1

两边除以

- 3

\frac{- 3 u ^{2}}{- 3} = \frac{- 1}{- 3}

化简

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{1}{3}, u = - \frac{1}{3}

u = tan (x)

代回

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

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

tan (x) = \frac{1}{3} : x = arctan (\frac{1}{3}) + πn

tan (x) = \frac{1}{3}

使用反三角函数性质

tan (x) = \frac{1}{3}

tan (x) = \frac{1}{3}

的通解

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

x = arctan (\frac{1}{3}) + πn

x = arctan (\frac{1}{3}) + πn

tan (x) = - \frac{1}{3} : x = arctan (- \frac{1}{3}) + πn

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

使用反三角函数性质

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

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

的通解

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

x = arctan (- \frac{1}{3}) + πn

x = arctan (- \frac{1}{3}) + πn

合并所有解

x = arctan (\frac{1}{3}) + πn, x = arctan (- \frac{1}{3}) + πn

以小数形式表示解

x = 0.52359 \dots + πn, x = - 0.52359 \dots + πn

作图

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