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受欢迎的三角函数 >

3tan^4(θ)+1= 2/(tan^2(θ))

解答

3 tan^{4} (θ) + 1 = \frac{2}{tan ^{2} ( θ )}

解答

θ = 0.71287 \dots + πn, θ = - 0.71287 \dots + πn

+1

度数

θ = 40.84445 \dots^{\circ} + 18 0^{\circ} n, θ = - 40.84445 \dots^{\circ} + 18 0^{\circ} n

求解步骤

3 tan^{4} (θ) + 1 = \frac{2}{tan ^{2} ( θ )}

用替代法求解

3 tan^{4} (θ) + 1 = \frac{2}{tan ^{2} ( θ )}

令

: tan (θ) = u

3 u^{4} + 1 = \frac{2}{u ^{2}}

3 u^{4} + 1 = \frac{2}{u ^{2}} : u = 0.74741 \dots, u = - 0.74741 \dots

3 u^{4} + 1 = \frac{2}{u ^{2}}

在两边乘以

u^{2}

3 u^{4} + 1 = \frac{2}{u ^{2}}

在两边乘以

u^{2}

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

化简

3 u^{4} u^{2} : 3 u^{6}

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

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

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

= 3 u^{4 + 2}

数字相加:

4 + 2 = 6

= 3 u^{6}

3 u^{6} + u^{2} = 2

3 u^{6} + u^{2} = 2

解

3 u^{6} + u^{2} = 2 : u = 0.74741 \dots, u = - 0.74741 \dots

3 u^{6} + u^{2} = 2

将

2

para o lado esquerdo

3 u^{6} + u^{2} = 2

两边减去

2

3 u^{6} + u^{2} - 2 = 2 - 2

化简

3 u^{6} + u^{2} - 2 = 0

3 u^{6} + u^{2} - 2 = 0

用

v = u^{2}

和

v^{3} = u^{6}

改写方程式

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

解

3 v^{3} + v - 2 = 0 : v \approx 0.74741 \dots

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

使用牛顿-拉弗森方法找到

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

的一个解:

v \approx 0.74741 \dots

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

牛顿-拉弗森近似法定义

f (v) = 3 v^{3} + v - 2

找到

f^{^{'}} (v) : 9 v^{2} + 1

\frac{d}{d v} (3 v^{3} + v - 2)

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d v} (3 v^{3}) + \frac{d v}{d v} - \frac{d}{d v} (2)

\frac{d}{d v} (3 v^{3}) = 9 v^{2}

\frac{d}{d v} (3 v^{3})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 3 \frac{d}{d v} (v^{3})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3 \cdot 3 v^{3 - 1}

化简

= 9 v^{2}

\frac{d v}{d v} = 1

\frac{d v}{d v}

使用常见微分定则:

\frac{d v}{d v} = 1

= 1

\frac{d}{d v} (2) = 0

\frac{d}{d v} (2)

常数微分:

\frac{d}{d x} (a) = 0

= 0

= 9 v^{2} + 1 - 0

化简

= 9 v^{2} + 1

令

v_{0} = 2

计算

v_{n + 1}

至

Δ v_{n + 1} < 0.000001

v_{1} = 1.35135 \dots : Δ v_{1} = 0.64864 \dots

f (v_{0}) = 3 \cdot 2^{3} + 2 - 2 = 24

f^{^{'}} (v_{0}) = 9 \cdot 2^{2} + 1 = 37

v_{1} = 1.35135 \dots

Δ v_{1} = ∣ 1.35135 \dots - 2 ∣ = 0.64864 \dots

Δ v_{1} = 0.64864 \dots

v_{2} = 0.96393 \dots : Δ v_{2} = 0.38741 \dots

f (v_{1}) = 3 \cdot 1.35135 \dots^{3} + 1.35135 \dots - 2 = 6.75466 \dots

f^{^{'}} (v_{1}) = 9 \cdot 1.35135 \dots^{2} + 1 = 17.43535 \dots

v_{2} = 0.96393 \dots

Δ v_{2} = ∣ 0.96393 \dots - 1.35135 \dots ∣ = 0.38741 \dots

Δ v_{2} = 0.38741 \dots

v_{3} = 0.78760 \dots : Δ v_{3} = 0.17633 \dots

f (v_{2}) = 3 \cdot 0.96393 \dots^{3} + 0.96393 \dots - 2 = 1.65095 \dots

f^{^{'}} (v_{2}) = 9 \cdot 0.96393 \dots^{2} + 1 = 9.36261 \dots

v_{3} = 0.78760 \dots

Δ v_{3} = ∣ 0.78760 \dots - 0.96393 \dots ∣ = 0.17633 \dots

Δ v_{3} = 0.17633 \dots

v_{4} = 0.74912 \dots : Δ v_{4} = 0.03847 \dots

f (v_{3}) = 3 \cdot 0.78760 \dots^{3} + 0.78760 \dots - 2 = 0.25330 \dots

f^{^{'}} (v_{3}) = 9 \cdot 0.78760 \dots^{2} + 1 = 6.58288 \dots

v_{4} = 0.74912 \dots

Δ v_{4} = ∣ 0.74912 \dots - 0.78760 \dots ∣ = 0.03847 \dots

Δ v_{4} = 0.03847 \dots

v_{5} = 0.74741 \dots : Δ v_{5} = 0.00170 \dots

f (v_{4}) = 3 \cdot 0.74912 \dots^{3} + 0.74912 \dots - 2 = 0.01032 \dots

f^{^{'}} (v_{4}) = 9 \cdot 0.74912 \dots^{2} + 1 = 6.05069 \dots

v_{5} = 0.74741 \dots

Δ v_{5} = ∣ 0.74741 \dots - 0.74912 \dots ∣ = 0.00170 \dots

Δ v_{5} = 0.00170 \dots

v_{6} = 0.74741 \dots : Δ v_{6} = 3.25433 E - 6

f (v_{5}) = 3 \cdot 0.74741 \dots^{3} + 0.74741 \dots - 2 = 0.00001 \dots

f^{^{'}} (v_{5}) = 9 \cdot 0.74741 \dots^{2} + 1 = 6.02770 \dots

v_{6} = 0.74741 \dots

Δ v_{6} = ∣ 0.74741 \dots - 0.74741 \dots ∣ = 3.25433 E - 6

Δ v_{6} = 3.25433 E - 6

v_{7} = 0.74741 \dots : Δ v_{7} = 1.1819 E - 11

f (v_{6}) = 3 \cdot 0.74741 \dots^{3} + 0.74741 \dots - 2 = 7.12408 E - 11

f^{^{'}} (v_{6}) = 9 \cdot 0.74741 \dots^{2} + 1 = 6.02766 \dots

v_{7} = 0.74741 \dots

Δ v_{7} = ∣ 0.74741 \dots - 0.74741 \dots ∣ = 1.1819 E - 11

Δ v_{7} = 1.1819 E - 11

v \approx 0.74741 \dots

使用长除法

Eq u a t i o n 0 : \frac{3 v ^{3} + v - 2}{v - 0.74741 \dots} = 3 v^{2} + 2.24224 \dots v + 2.67588 \dots

3 v^{2} + 2.24224 \dots v + 2.67588 \dots \approx 0

使用牛顿-拉弗森方法找到

3 v^{2} + 2.24224 \dots v + 2.67588 \dots = 0

的一个解:

v \in R

无解

3 v^{2} + 2.24224 \dots v + 2.67588 \dots = 0

牛顿-拉弗森近似法定义

f (v) = 3 v^{2} + 2.24224 \dots v + 2.67588 \dots

找到

f^{^{'}} (v) : 6 v + 2.24224 \dots

\frac{d}{d v} (3 v^{2} + 2.24224 \dots v + 2.67588 \dots)

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d v} (3 v^{2}) + \frac{d}{d v} (2.24224 \dots v) + \frac{d}{d v} (2.67588 \dots)

\frac{d}{d v} (3 v^{2}) = 6 v

\frac{d}{d v} (3 v^{2})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 3 \frac{d}{d v} (v^{2})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3 \cdot 2 v^{2 - 1}

化简

= 6 v

\frac{d}{d v} (2.24224 \dots v) = 2.24224 \dots

\frac{d}{d v} (2.24224 \dots v)

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 2.24224 \dots \frac{d v}{d v}

使用常见微分定则:

\frac{d v}{d v} = 1

= 2.24224 \dots \cdot 1

化简

= 2.24224 \dots

\frac{d}{d v} (2.67588 \dots) = 0

\frac{d}{d v} (2.67588 \dots)

常数微分:

\frac{d}{d x} (a) = 0

= 0

= 6 v + 2.24224 \dots + 0

化简

= 6 v + 2.24224 \dots

令

v_{0} = - 1

计算

v_{n + 1}

至

Δ v_{n + 1} < 0.000001

v_{1} = - 0.08625 \dots : Δ v_{1} = 0.91374 \dots

f (v_{0}) = 3 (- 1)^{2} + 2.24224 \dots (- 1) + 2.67588 \dots = 3.43364 \dots

f^{^{'}} (v_{0}) = 6 (- 1) + 2.24224 \dots = - 3.75775 \dots

v_{1} = - 0.08625 \dots

Δ v_{1} = ∣ - 0.08625 \dots - (- 1) ∣ = 0.91374 \dots

Δ v_{1} = 0.91374 \dots

v_{2} = - 1.53853 \dots : Δ v_{2} = 1.45228 \dots

f (v_{1}) = 3 (- 0.08625 \dots)^{2} + 2.24224 \dots (- 0.08625 \dots) + 2.67588 \dots = 2.50480 \dots

f^{^{'}} (v_{1}) = 6 (- 0.08625 \dots) + 2.24224 \dots = 1.72473 \dots

v_{2} = - 1.53853 \dots

Δ v_{2} = ∣ - 1.53853 \dots - (- 0.08625 \dots) ∣ = 1.45228 \dots

Δ v_{2} = 1.45228 \dots

v_{3} = - 0.63319 \dots : Δ v_{3} = 0.90533 \dots

f (v_{2}) = 3 (- 1.53853 \dots)^{2} + 2.24224 \dots (- 1.53853 \dots) + 2.67588 \dots = 6.32739 \dots

f^{^{'}} (v_{2}) = 6 (- 1.53853 \dots) + 2.24224 \dots = - 6.98896 \dots

v_{3} = - 0.63319 \dots

Δ v_{3} = ∣ - 0.63319 \dots - (- 1.53853 \dots) ∣ = 0.90533 \dots

Δ v_{3} = 0.90533 \dots

v_{4} = 0.94614 \dots : Δ v_{4} = 1.57933 \dots

f (v_{3}) = 3 (- 0.63319 \dots)^{2} + 2.24224 \dots (- 0.63319 \dots) + 2.67588 \dots = 2.45891 \dots

f^{^{'}} (v_{3}) = 6 (- 0.63319 \dots) + 2.24224 \dots = - 1.55693 \dots

v_{4} = 0.94614 \dots

Δ v_{4} = ∣ 0.94614 \dots - (- 0.63319 \dots) ∣ = 1.57933 \dots

Δ v_{4} = 1.57933 \dots

v_{5} = 0.00121 \dots : Δ v_{5} = 0.94492 \dots

f (v_{4}) = 3 \cdot 0.94614 \dots^{2} + 2.24224 \dots \cdot 0.94614 \dots + 2.67588 \dots = 7.48291 \dots

f^{^{'}} (v_{4}) = 6 \cdot 0.94614 \dots + 2.24224 \dots = 7.91909 \dots

v_{5} = 0.00121 \dots

Δ v_{5} = ∣ 0.00121 \dots - 0.94614 \dots ∣ = 0.94492 \dots

Δ v_{5} = 0.94492 \dots

v_{6} = - 1.18951 \dots : Δ v_{6} = 1.19073 \dots

f (v_{5}) = 3 \cdot 0.00121 \dots^{2} + 2.24224 \dots \cdot 0.00121 \dots + 2.67588 \dots = 2.67862 \dots

f^{^{'}} (v_{5}) = 6 \cdot 0.00121 \dots + 2.24224 \dots = 2.24956 \dots

v_{6} = - 1.18951 \dots

Δ v_{6} = ∣ - 1.18951 \dots - 0.00121 \dots ∣ = 1.19073 \dots

Δ v_{6} = 1.19073 \dots

v_{7} = - 0.32052 \dots : Δ v_{7} = 0.86898 \dots

f (v_{6}) = 3 (- 1.18951 \dots)^{2} + 2.24224 \dots (- 1.18951 \dots) + 2.67588 \dots = 4.25353 \dots

f^{^{'}} (v_{6}) = 6 (- 1.18951 \dots) + 2.24224 \dots = - 4.89483 \dots

v_{7} = - 0.32052 \dots

Δ v_{7} = ∣ - 0.32052 \dots - (- 1.18951 \dots) ∣ = 0.86898 \dots

Δ v_{7} = 0.86898 \dots

v_{8} = - 7.42043 \dots : Δ v_{8} = 7.09990 \dots

f (v_{7}) = 3 (- 0.32052 \dots)^{2} + 2.24224 \dots (- 0.32052 \dots) + 2.67588 \dots = 2.26540 \dots

f^{^{'}} (v_{7}) = 6 (- 0.32052 \dots) + 2.24224 \dots = 0.31907 \dots

v_{8} = - 7.42043 \dots

Δ v_{8} = ∣ - 7.42043 \dots - (- 0.32052 \dots) ∣ = 7.09990 \dots

Δ v_{8} = 7.09990 \dots

无法得出解

解是

v \approx 0.74741 \dots

v \approx 0.74741 \dots

代回

v = u^{2}

，求解

u

解

u^{2} = 0.74741 \dots : u = 0.74741 \dots, u = - 0.74741 \dots

u^{2} = 0.74741 \dots

对于

x^{2} = f (a)

解为

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

u = 0.74741 \dots, u = - 0.74741 \dots

解为

u = 0.74741 \dots, u = - 0.74741 \dots

u = 0.74741 \dots, u = - 0.74741 \dots

验证解

找到无定义的点（奇点）:

u = 0

取

\frac{2}{u ^{2}}

的分母，令其等于零

解

u^{2} = 0 : u = 0

u^{2} = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = 0.74741 \dots, u = - 0.74741 \dots

u = tan (θ)

代回

tan (θ) = 0.74741 \dots, tan (θ) = - 0.74741 \dots

tan (θ) = 0.74741 \dots, tan (θ) = - 0.74741 \dots

tan (θ) = 0.74741 \dots : θ = arctan (0.74741 \dots) + πn

tan (θ) = 0.74741 \dots

使用反三角函数性质

tan (θ) = 0.74741 \dots

tan (θ) = 0.74741 \dots

的通解

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

θ = arctan (0.74741 \dots) + πn

θ = arctan (0.74741 \dots) + πn

tan (θ) = - 0.74741 \dots : θ = arctan (- 0.74741 \dots) + πn

tan (θ) = - 0.74741 \dots

使用反三角函数性质

tan (θ) = - 0.74741 \dots

tan (θ) = - 0.74741 \dots

的通解

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

θ = arctan (- 0.74741 \dots) + πn

θ = arctan (- 0.74741 \dots) + πn

合并所有解

θ = arctan (0.74741 \dots) + πn, θ = arctan (- 0.74741 \dots) + πn

以小数形式表示解

θ = 0.71287 \dots + πn, θ = - 0.71287 \dots + πn

作图

流行的例子

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