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受欢迎的三角函数 >

cos^2(a)= 2/(3sin(a))

解答

cos^{2} (a) = \frac{2}{3 sin ( a )}

解答

a \in R 无解

求解步骤

cos^{2} (a) = \frac{2}{3 sin ( a )}

两边减去

\frac{2}{3 sin ( a )}

cos^{2} (a) - \frac{2}{3 sin ( a )} = 0

化简

cos^{2} (a) - \frac{2}{3 sin ( a )} : \frac{3 cos ^{2} ( a ) sin ( a ) - 2}{3 sin ( a )}

cos^{2} (a) - \frac{2}{3 sin ( a )}

将项转换为分式:

cos^{2} (a) = \frac{cos ^{2} ( a ) 3 sin ( a )}{3 sin ( a )}

= \frac{cos ^{2} ( a ) \cdot 3 sin ( a )}{3 sin ( a )} - \frac{2}{3 sin ( a )}

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{cos ^{2} ( a ) \cdot 3 sin ( a ) - 2}{3 sin ( a )}

\frac{3 cos ^{2} ( a ) sin ( a ) - 2}{3 sin ( a )} = 0

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

3 cos^{2} (a) sin (a) - 2 = 0

使用三角恒等式改写

- 2 + 3 cos^{2} (a) sin (a)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= - 2 + 3 (1 - sin^{2} (a)) sin (a)

- 2 + (1 - sin^{2} (a)) \cdot 3 sin (a) = 0

用替代法求解

- 2 + (1 - sin^{2} (a)) \cdot 3 sin (a) = 0

令

: sin (a) = u

- 2 + (1 - u^{2}) \cdot 3 u = 0

- 2 + (1 - u^{2}) \cdot 3 u = 0 : u \approx - 1.24001 \dots

- 2 + (1 - u^{2}) \cdot 3 u = 0

展开

- 2 + (1 - u^{2}) \cdot 3 u : - 2 + 3 u - 3 u^{3}

- 2 + (1 - u^{2}) \cdot 3 u

= - 2 + 3 u (1 - u^{2})

乘开

3 u (1 - u^{2}) : 3 u - 3 u^{3}

3 u (1 - u^{2})

使用分配律:

a (b - c) = ab - a c

a = 3 u, b = 1, c = u^{2}

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

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

化简

3 \cdot 1 \cdot u - 3 u^{2} u : 3 u - 3 u^{3}

3 \cdot 1 \cdot u - 3 u^{2} u

3 \cdot 1 \cdot u = 3 u

3 \cdot 1 \cdot u

数字相乘:

3 \cdot 1 = 3

= 3 u

3 u^{2} u = 3 u^{3}

3 u^{2} u

使用指数法则:

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

u^{2} u = u^{2 + 1}

= 3 u^{2 + 1}

数字相加:

2 + 1 = 3

= 3 u^{3}

= 3 u - 3 u^{3}

= 3 u - 3 u^{3}

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

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

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a = 0

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

使用牛顿-拉弗森方法找到

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

的一个解:

u \approx - 1.24001 \dots

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

牛顿-拉弗森近似法定义

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

找到

f^{^{'}} (u) : - 9 u^{2} + 3

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

使用微分加减法定则:

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

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

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

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

将常数提出:

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

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

使用幂法则:

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

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

化简

= 9 u^{2}

\frac{d}{d u} (3 u) = 3

\frac{d}{d u} (3 u)

将常数提出:

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

= 3 \frac{d u}{d u}

使用常见微分定则:

\frac{d u}{d u} = 1

= 3 \cdot 1

化简

= 3

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

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

常数微分:

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

= 0

= - 9 u^{2} + 3 - 0

化简

= - 9 u^{2} + 3

令

u_{0} = - 1

计算

u_{n + 1}

至

Δ u_{n + 1} < 0.000001

u_{1} = - 1.33333 \dots : Δ u_{1} = 0.33333 \dots

f (u_{0}) = - 3 (- 1)^{3} + 3 (- 1) - 2 = - 2

f^{^{'}} (u_{0}) = - 9 (- 1)^{2} + 3 = - 6

u_{1} = - 1.33333 \dots

Δ u_{1} = ∣ - 1.33333 \dots - (- 1) ∣ = 0.33333 \dots

Δ u_{1} = 0.33333 \dots

u_{2} = - 1.24786 \dots : Δ u_{2} = 0.08547 \dots

f (u_{1}) = - 3 (- 1.33333 \dots)^{3} + 3 (- 1.33333 \dots) - 2 = 1.11111 \dots

f^{^{'}} (u_{1}) = - 9 (- 1.33333 \dots)^{2} + 3 = - 13

u_{2} = - 1.24786 \dots

Δ u_{2} = ∣ - 1.24786 \dots - (- 1.33333 \dots) ∣ = 0.08547 \dots

Δ u_{2} = 0.08547 \dots

u_{3} = - 1.24007 \dots : Δ u_{3} = 0.00778 \dots

f (u_{2}) = - 3 (- 1.24786 \dots)^{3} + 3 (- 1.24786 \dots) - 2 = 0.08578 \dots

f^{^{'}} (u_{2}) = - 9 (- 1.24786 \dots)^{2} + 3 = - 11.01446 \dots

u_{3} = - 1.24007 \dots

Δ u_{3} = ∣ - 1.24007 \dots - (- 1.24786 \dots) ∣ = 0.00778 \dots

Δ u_{3} = 0.00778 \dots

u_{4} = - 1.24001 \dots : Δ u_{4} = 0.00006 \dots

f (u_{3}) = - 3 (- 1.24007 \dots)^{3} + 3 (- 1.24007 \dots) - 2 = 0.00067 \dots

f^{^{'}} (u_{3}) = - 9 (- 1.24007 \dots)^{2} + 3 = - 10.84006 \dots

u_{4} = - 1.24001 \dots

Δ u_{4} = ∣ - 1.24001 \dots - (- 1.24007 \dots) ∣ = 0.00006 \dots

Δ u_{4} = 0.00006 \dots

u_{5} = - 1.24001 \dots : Δ u_{5} = 4.05057 E - 9

f (u_{4}) = - 3 (- 1.24001 \dots)^{3} + 3 (- 1.24001 \dots) - 2 = 4.39028 E - 8

f^{^{'}} (u_{4}) = - 9 (- 1.24001 \dots)^{2} + 3 = - 10.83866 \dots

u_{5} = - 1.24001 \dots

Δ u_{5} = ∣ - 1.24001 \dots - (- 1.24001 \dots) ∣ = 4.05057 E - 9

Δ u_{5} = 4.05057 E - 9

u \approx - 1.24001 \dots

使用长除法

Eq u a t i o n 0 : \frac{- 3 u ^{3} + 3 u - 2}{u + 1.24001 \dots} = - 3 u^{2} + 3.72003 \dots u - 1.61288 \dots

- 3 u^{2} + 3.72003 \dots u - 1.61288 \dots \approx 0

使用牛顿-拉弗森方法找到

- 3 u^{2} + 3.72003 \dots u - 1.61288 \dots = 0

的一个解:

u \in R

无解

- 3 u^{2} + 3.72003 \dots u - 1.61288 \dots = 0

牛顿-拉弗森近似法定义

f (u) = - 3 u^{2} + 3.72003 \dots u - 1.61288 \dots

找到

f^{^{'}} (u) : - 6 u + 3.72003 \dots

\frac{d}{d u} (- 3 u^{2} + 3.72003 \dots u - 1.61288 \dots)

使用微分加减法定则:

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

= - \frac{d}{d u} (3 u^{2}) + \frac{d}{d u} (3.72003 \dots u) - \frac{d}{d u} (1.61288 \dots)

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

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

将常数提出:

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

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

使用幂法则:

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

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

化简

= 6 u

\frac{d}{d u} (3.72003 \dots u) = 3.72003 \dots

\frac{d}{d u} (3.72003 \dots u)

将常数提出:

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

= 3.72003 \dots \frac{d u}{d u}

使用常见微分定则:

\frac{d u}{d u} = 1

= 3.72003 \dots \cdot 1

化简

= 3.72003 \dots

\frac{d}{d u} (1.61288 \dots) = 0

\frac{d}{d u} (1.61288 \dots)

常数微分:

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

= 0

= - 6 u + 3.72003 \dots - 0

化简

= - 6 u + 3.72003 \dots

令

u_{0} = 0

计算

u_{n + 1}

至

Δ u_{n + 1} < 0.000001

u_{1} = 0.43356 \dots : Δ u_{1} = 0.43356 \dots

f (u_{0}) = - 3 \cdot 0^{2} + 3.72003 \dots \cdot 0 - 1.61288 \dots = - 1.61288 \dots

f^{^{'}} (u_{0}) = - 6 \cdot 0 + 3.72003 \dots = 3.72003 \dots

u_{1} = 0.43356 \dots

Δ u_{1} = ∣ 0.43356 \dots - 0 ∣ = 0.43356 \dots

Δ u_{1} = 0.43356 \dots

u_{2} = 0.93770 \dots : Δ u_{2} = 0.50413 \dots

f (u_{1}) = - 3 \cdot 0.43356 \dots^{2} + 3.72003 \dots \cdot 0.43356 \dots - 1.61288 \dots = - 0.56394 \dots

f^{^{'}} (u_{1}) = - 6 \cdot 0.43356 \dots + 3.72003 \dots = 1.11862 \dots

u_{2} = 0.93770 \dots

Δ u_{2} = ∣ 0.93770 \dots - 0.43356 \dots ∣ = 0.50413 \dots

Δ u_{2} = 0.50413 \dots

u_{3} = 0.53771 \dots : Δ u_{3} = 0.39999 \dots

f (u_{2}) = - 3 \cdot 0.93770 \dots^{2} + 3.72003 \dots \cdot 0.93770 \dots - 1.61288 \dots = - 0.76246 \dots

f^{^{'}} (u_{2}) = - 6 \cdot 0.93770 \dots + 3.72003 \dots = - 1.90620 \dots

u_{3} = 0.53771 \dots

Δ u_{3} = ∣ 0.53771 \dots - 0.93770 \dots ∣ = 0.39999 \dots

Δ u_{3} = 0.39999 \dots

u_{4} = 1.50982 \dots : Δ u_{4} = 0.97210 \dots

f (u_{3}) = - 3 \cdot 0.53771 \dots^{2} + 3.72003 \dots \cdot 0.53771 \dots - 1.61288 \dots = - 0.47998 \dots

f^{^{'}} (u_{3}) = - 6 \cdot 0.53771 \dots + 3.72003 \dots = 0.49375 \dots

u_{4} = 1.50982 \dots

Δ u_{4} = ∣ 1.50982 \dots - 0.53771 \dots ∣ = 0.97210 \dots

Δ u_{4} = 0.97210 \dots

u_{5} = 0.97881 \dots : Δ u_{5} = 0.53100 \dots

f (u_{4}) = - 3 \cdot 1.50982 \dots^{2} + 3.72003 \dots \cdot 1.50982 \dots - 1.61288 \dots = - 2.83498 \dots

f^{^{'}} (u_{4}) = - 6 \cdot 1.50982 \dots + 3.72003 \dots = - 5.33890 \dots

u_{5} = 0.97881 \dots

Δ u_{5} = ∣ 0.97881 \dots - 1.50982 \dots ∣ = 0.53100 \dots

Δ u_{5} = 0.53100 \dots

u_{6} = 0.58589 \dots : Δ u_{6} = 0.39291 \dots

f (u_{5}) = - 3 \cdot 0.97881 \dots^{2} + 3.72003 \dots \cdot 0.97881 \dots - 1.61288 \dots = - 0.84590 \dots

f^{^{'}} (u_{5}) = - 6 \cdot 0.97881 \dots + 3.72003 \dots = - 2.15286 \dots

u_{6} = 0.58589 \dots

Δ u_{6} = ∣ 0.58589 \dots - 0.97881 \dots ∣ = 0.39291 \dots

Δ u_{6} = 0.39291 \dots

u_{7} = 2.84908 \dots : Δ u_{7} = 2.26318 \dots

f (u_{6}) = - 3 \cdot 0.58589 \dots^{2} + 3.72003 \dots \cdot 0.58589 \dots - 1.61288 \dots = - 0.46315 \dots

f^{^{'}} (u_{6}) = - 6 \cdot 0.58589 \dots + 3.72003 \dots = 0.20464 \dots

u_{7} = 2.84908 \dots

Δ u_{7} = ∣ 2.84908 \dots - 0.58589 \dots ∣ = 2.26318 \dots

Δ u_{7} = 2.26318 \dots

u_{8} = 1.70017 \dots : Δ u_{8} = 1.14890 \dots

f (u_{7}) = - 3 \cdot 2.84908 \dots^{2} + 3.72003 \dots \cdot 2.84908 \dots - 1.61288 \dots = - 15.36607 \dots

f^{^{'}} (u_{7}) = - 6 \cdot 2.84908 \dots + 3.72003 \dots = - 13.37448 \dots

u_{8} = 1.70017 \dots

Δ u_{8} = ∣ 1.70017 \dots - 2.84908 \dots ∣ = 1.14890 \dots

Δ u_{8} = 1.14890 \dots

u_{9} = 1.08916 \dots : Δ u_{9} = 0.61101 \dots

f (u_{8}) = - 3 \cdot 1.70017 \dots^{2} + 3.72003 \dots \cdot 1.70017 \dots - 1.61288 \dots = - 3.95997 \dots

f^{^{'}} (u_{8}) = - 6 \cdot 1.70017 \dots + 3.72003 \dots = - 6.48103 \dots

u_{9} = 1.08916 \dots

Δ u_{9} = ∣ 1.08916 \dots - 1.70017 \dots ∣ = 0.61101 \dots

Δ u_{9} = 0.61101 \dots

u_{10} = 0.69129 \dots : Δ u_{10} = 0.39787 \dots

f (u_{9}) = - 3 \cdot 1.08916 \dots^{2} + 3.72003 \dots \cdot 1.08916 \dots - 1.61288 \dots = - 1.12000 \dots

f^{^{'}} (u_{9}) = - 6 \cdot 1.08916 \dots + 3.72003 \dots = - 2.81496 \dots

u_{10} = 0.69129 \dots

Δ u_{10} = ∣ 0.69129 \dots - 1.08916 \dots ∣ = 0.39787 \dots

Δ u_{10} = 0.39787 \dots

u_{11} = - 0.41903 \dots : Δ u_{11} = 1.11032 \dots

f (u_{10}) = - 3 \cdot 0.69129 \dots^{2} + 3.72003 \dots \cdot 0.69129 \dots - 1.61288 \dots = - 0.47491 \dots

f^{^{'}} (u_{10}) = - 6 \cdot 0.69129 \dots + 3.72003 \dots = - 0.42772 \dots

u_{11} = - 0.41903 \dots

Δ u_{11} = ∣ - 0.41903 \dots - 0.69129 \dots ∣ = 1.11032 \dots

Δ u_{11} = 1.11032 \dots

无法得出解

解是

u \approx - 1.24001 \dots

u = sin (a)

代回

sin (a) \approx - 1.24001 \dots

sin (a) \approx - 1.24001 \dots

sin (a) = - 1.24001 \dots :

无解

sin (a) = - 1.24001 \dots

- 1 \leq sin (x) \leq 1

无解

合并所有解

a \in R 无解

作图

流行的例子

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

tan^2(x)+cos^2(x)-1=0

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

cot^3(x)+cot(x)=0

cot^{3} (x) + cot (x) = 0