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受欢迎的三角函数 >

(sin^2(x))/(cos(x))=16.33

解答

\frac{sin ^{2} ( x )}{cos ( x )} = 16.33

解答

x = 1.50974 \dots + 2 πn, x = 2 π - 1.50974 \dots + 2 πn

+1

度数

x = 86.50226 \dots^{\circ} + 36 0^{\circ} n, x = 273.49773 \dots^{\circ} + 36 0^{\circ} n

求解步骤

\frac{sin ^{2} ( x )}{cos ( x )} = 16.33

两边减去

16.33

\frac{sin ^{2} ( x )}{cos ( x )} - 16.33 = 0

使用三角恒等式改写

- 16.33 + \frac{sin ^{2} ( x )}{cos ( x )}

使用毕达哥拉斯恒等式:

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

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

= - 16.33 + \frac{1 - cos ^{2} ( x )}{cos ( x )}

- 16.33 + \frac{1 - cos ^{2} ( x )}{cos ( x )} = 0

用替代法求解

- 16.33 + \frac{1 - cos ^{2} ( x )}{cos ( x )} = 0

令

: cos (x) = u

- 16.33 + \frac{1 - u ^{2}}{u} = 0

- 16.33 + \frac{1 - u ^{2}}{u} = 0 : u = - \frac{1633 + 2706689}{200}, u = \frac{2706689 - 1633}{200}

- 16.33 + \frac{1 - u ^{2}}{u} = 0

在两边乘以

u

- 16.33 + \frac{1 - u ^{2}}{u} = 0

在两边乘以

u

- 16.33 u + \frac{1 - u ^{2}}{u} u = 0 \cdot u

化简

- 16.33 u + \frac{1 - u ^{2}}{u} u = 0 \cdot u

化简

\frac{1 - u ^{2}}{u} u : 1 - u^{2}

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

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 1 - u ^{2} ) u}{u}

约分:

u

= 1 - u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

- 16.33 u + 1 - u^{2} = 0

- 16.33 u + 1 - u^{2} = 0

- 16.33 u + 1 - u^{2} = 0

解

- 16.33 u + 1 - u^{2} = 0 : u = - \frac{1633 + 2706689}{200}, u = \frac{2706689 - 1633}{200}

- 16.33 u + 1 - u^{2} = 0

在两边乘以

100

- 16.33 u + 1 - u^{2} = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

- 16.33 u \cdot 100 + 1 \cdot 100 - u^{2} \cdot 100 = 0 \cdot 100

整理后得

- 1633 u + 100 - 100 u^{2} = 0

- 1633 u + 100 - 100 u^{2} = 0

改写成标准形式

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

- 100 u^{2} - 1633 u + 100 = 0

使用求根公式求解

- 100 u^{2} - 1633 u + 100 = 0

二次方程求根公式:

若

a = - 100, b = - 1633, c = 100

u_{1, 2} = \frac{- ( - 1633 ) \pm ( - 1633 ) ^{2} - 4 ( - 100 ) \cdot 100}{2 ( - 100 )}

u_{1, 2} = \frac{- ( - 1633 ) \pm ( - 1633 ) ^{2} - 4 ( - 100 ) \cdot 100}{2 ( - 100 )}

(- 1633)^{2} - 4 (- 100) \cdot 100 = 2706689

(- 1633)^{2} - 4 (- 100) \cdot 100

使用法则

- (- a) = a

= (- 1633)^{2} + 4 \cdot 100 \cdot 100

使用指数法则:

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

若

n

是偶数

(- 1633)^{2} = 163 3^{2}

= 163 3^{2} + 4 \cdot 100 \cdot 100

数字相乘:

4 \cdot 100 \cdot 100 = 40000

= 163 3^{2} + 40000

163 3^{2} = 2666689

= 2666689 + 40000

数字相加:

2666689 + 40000 = 2706689

= 2706689

u_{1, 2} = \frac{- ( - 1633 ) \pm 2706689}{2 ( - 100 )}

将解分隔开

u_{1} = \frac{- ( - 1633 ) + 2706689}{2 ( - 100 )}, u_{2} = \frac{- ( - 1633 ) - 2706689}{2 ( - 100 )}

u = \frac{- ( - 1633 ) + 2706689}{2 ( - 100 )} : - \frac{1633 + 2706689}{200}

\frac{- ( - 1633 ) + 2706689}{2 ( - 100 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{1633 + 2706689}{- 2 \cdot 100}

数字相乘:

2 \cdot 100 = 200

= \frac{1633 + 2706689}{- 200}

使用分式法则:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{1633 + 2706689}{200}

u = \frac{- ( - 1633 ) - 2706689}{2 ( - 100 )} : \frac{2706689 - 1633}{200}

\frac{- ( - 1633 ) - 2706689}{2 ( - 100 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{1633 - 2706689}{- 2 \cdot 100}

数字相乘:

2 \cdot 100 = 200

= \frac{1633 - 2706689}{- 200}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

1633 - 2706689 = - (2706689 - 1633)

= \frac{2706689 - 1633}{200}

二次方程组的解是:

u = - \frac{1633 + 2706689}{200}, u = \frac{2706689 - 1633}{200}

u = - \frac{1633 + 2706689}{200}, u = \frac{2706689 - 1633}{200}

验证解

找到无定义的点（奇点）:

u = 0

取

- 16.33 + \frac{1 - u ^{2}}{u}

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = - \frac{1633 + 2706689}{200}, u = \frac{2706689 - 1633}{200}

u = cos (x)

代回

cos (x) = - \frac{1633 + 2706689}{200}, cos (x) = \frac{2706689 - 1633}{200}

cos (x) = - \frac{1633 + 2706689}{200}, cos (x) = \frac{2706689 - 1633}{200}

cos (x) = - \frac{1633 + 2706689}{200} :

无解

cos (x) = - \frac{1633 + 2706689}{200}

- 1 \leq cos (x) \leq 1

无解

cos (x) = \frac{2706689 - 1633}{200} : x = arccos (\frac{2706689 - 1633}{200}) + 2 πn, x = 2 π - arccos (\frac{2706689 - 1633}{200}) + 2 πn

cos (x) = \frac{2706689 - 1633}{200}

使用反三角函数性质

cos (x) = \frac{2706689 - 1633}{200}

cos (x) = \frac{2706689 - 1633}{200}

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{2706689 - 1633}{200}) + 2 πn, x = 2 π - arccos (\frac{2706689 - 1633}{200}) + 2 πn

x = arccos (\frac{2706689 - 1633}{200}) + 2 πn, x = 2 π - arccos (\frac{2706689 - 1633}{200}) + 2 πn

合并所有解

x = arccos (\frac{2706689 - 1633}{200}) + 2 πn, x = 2 π - arccos (\frac{2706689 - 1633}{200}) + 2 πn

以小数形式表示解

x = 1.50974 \dots + 2 πn, x = 2 π - 1.50974 \dots + 2 πn

作图

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