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受欢迎的三角函数 >

1-tan^2(y)=2sec(y)-4

解答

1 - tan^{2} (y) = 2 sec (y) - 4

解答

y = 1.84864 \dots + 2 πn, y = - 1.84864 \dots + 2 πn, y = 0.91772 \dots + 2 πn, y = 2 π - 0.91772 \dots + 2 πn

+1

度数

y = 105.91981 \dots^{\circ} + 36 0^{\circ} n, y = - 105.91981 \dots^{\circ} + 36 0^{\circ} n, y = 52.58201 \dots^{\circ} + 36 0^{\circ} n, y = 307.41798 \dots^{\circ} + 36 0^{\circ} n

求解步骤

1 - tan^{2} (y) = 2 sec (y) - 4

两边减去

2 sec (y) - 4

- tan^{2} (y) - 2 sec (y) + 5 = 0

使用三角恒等式改写

5 - tan^{2} (y) - 2 sec (y)

使用毕达哥拉斯恒等式:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= 5 - (sec^{2} (y) - 1) - 2 sec (y)

化简

5 - (sec^{2} (y) - 1) - 2 sec (y) : - sec^{2} (y) - 2 sec (y) + 6

5 - (sec^{2} (y) - 1) - 2 sec (y)

- (sec^{2} (y) - 1) : - sec^{2} (y) + 1

- (sec^{2} (y) - 1)

打开括号

= - (sec^{2} (y)) - (- 1)

使用加减运算法则

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

= - sec^{2} (y) + 1

= 5 - sec^{2} (y) + 1 - 2 sec (y)

化简

5 - sec^{2} (y) + 1 - 2 sec (y) : - sec^{2} (y) - 2 sec (y) + 6

5 - sec^{2} (y) + 1 - 2 sec (y)

对同类项分组

= - sec^{2} (y) - 2 sec (y) + 5 + 1

数字相加:

5 + 1 = 6

= - sec^{2} (y) - 2 sec (y) + 6

= - sec^{2} (y) - 2 sec (y) + 6

= - sec^{2} (y) - 2 sec (y) + 6

6 - sec^{2} (y) - 2 sec (y) = 0

用替代法求解

6 - sec^{2} (y) - 2 sec (y) = 0

令

: sec (y) = u

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

6 - u^{2} - 2 u = 0 : u = - 1 - 7, u = 7 - 1

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 1, b = - 2, c = 6

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 1 ) \cdot 6}{2 ( - 1 )}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 1 ) \cdot 6}{2 ( - 1 )}

(- 2)^{2} - 4 (- 1) \cdot 6 = 27

(- 2)^{2} - 4 (- 1) \cdot 6

使用法则

- (- a) = a

= (- 2)^{2} + 4 \cdot 1 \cdot 6

使用指数法则:

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

若

n

是偶数

(- 2)^{2} = 2^{2}

= 2^{2} + 4 \cdot 1 \cdot 6

数字相乘:

4 \cdot 1 \cdot 6 = 24

= 2^{2} + 24

2^{2} = 4

= 4 + 24

数字相加:

4 + 24 = 28

= 28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

n ab = n a n b

= 7 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 27

u_{1, 2} = \frac{- ( - 2 ) \pm 2 7}{2 ( - 1 )}

将解分隔开

u_{1} = \frac{- ( - 2 ) + 2 7}{2 ( - 1 )}, u_{2} = \frac{- ( - 2 ) - 2 7}{2 ( - 1 )}

u = \frac{- ( - 2 ) + 2 7}{2 ( - 1 )} : - 1 - 7

\frac{- ( - 2 ) + 2 7}{2 ( - 1 )}

去除括号:

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

= \frac{2 + 2 7}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{2 + 2 7}{- 2}

使用分式法则:

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

= - \frac{2 + 2 7}{2}

消掉

\frac{2 + 2 7}{2} : 1 + 7

\frac{2 + 2 7}{2}

分解

2 + 27 : 2 (1 + 7)

2 + 27

改写为

= 2 \cdot 1 + 27

因式分解出通项

2

= 2 (1 + 7)

= \frac{2 ( 1 + 7 )}{2}

数字相除:

\frac{2}{2} = 1

= 1 + 7

= - (1 + 7)

打开括号

= - (1) - (7)

使用加减运算法则

+ (- a) = - a

= - 1 - 7

u = \frac{- ( - 2 ) - 2 7}{2 ( - 1 )} : 7 - 1

\frac{- ( - 2 ) - 2 7}{2 ( - 1 )}

去除括号:

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

= \frac{2 - 2 7}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{2 - 2 7}{- 2}

使用分式法则:

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

2 - 27 = - (27 - 2)

= \frac{2 7 - 2}{2}

分解

27 - 2 : 2 (7 - 1)

27 - 2

改写为

= 27 - 2 \cdot 1

因式分解出通项

2

= 2 (7 - 1)

= \frac{2 ( 7 - 1 )}{2}

数字相除:

\frac{2}{2} = 1

= 7 - 1

二次方程组的解是:

u = - 1 - 7, u = 7 - 1

u = sec (y)

代回

sec (y) = - 1 - 7, sec (y) = 7 - 1

sec (y) = - 1 - 7, sec (y) = 7 - 1

sec (y) = - 1 - 7 : y = arcsec (- 1 - 7) + 2 πn, y = - arcsec (- 1 - 7) + 2 πn

sec (y) = - 1 - 7

使用反三角函数性质

sec (y) = - 1 - 7

sec (y) = - 1 - 7

的通解

sec (x) = - a \Rightarrow x = arcsec (- a) + 2 πn, x = - arcsec (- a) + 2 πn

y = arcsec (- 1 - 7) + 2 πn, y = - arcsec (- 1 - 7) + 2 πn

y = arcsec (- 1 - 7) + 2 πn, y = - arcsec (- 1 - 7) + 2 πn

sec (y) = 7 - 1 : y = arcsec (7 - 1) + 2 πn, y = 2 π - arcsec (7 - 1) + 2 πn

sec (y) = 7 - 1

使用反三角函数性质

sec (y) = 7 - 1

sec (y) = 7 - 1

的通解

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

y = arcsec (7 - 1) + 2 πn, y = 2 π - arcsec (7 - 1) + 2 πn

y = arcsec (7 - 1) + 2 πn, y = 2 π - arcsec (7 - 1) + 2 πn

合并所有解

y = arcsec (- 1 - 7) + 2 πn, y = - arcsec (- 1 - 7) + 2 πn, y = arcsec (7 - 1) + 2 πn, y = 2 π - arcsec (7 - 1) + 2 πn

以小数形式表示解

y = 1.84864 \dots + 2 πn, y = - 1.84864 \dots + 2 πn, y = 0.91772 \dots + 2 πn, y = 2 π - 0.91772 \dots + 2 πn

作图

流行的例子

3 cot (x) + 4 = 7

cot^2(x)csc^2(x)+2csc^2(x)-cot^2(x)=2

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x) - cot^{2} (x) = 2

cos(t)= 3/5 ,0<t< pi/2

cos (t) = \frac{3}{5}, 0 < t < \frac{π}{2}

cos (x) = \frac{- 2}{7}

csc(x)= 9/(sqrt(17))

csc (x) = \frac{9}{17}