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受欢迎的三角函数 >

3cos(x)=sec(x)

解答

3 cos (x) = sec (x)

解答

x = 0.95531 \dots + 2 πn, x = 2 π - 0.95531 \dots + 2 πn, x = 2.18627 \dots + 2 πn, x = - 2.18627 \dots + 2 πn

+1

度数

x = 54.73561 \dots^{\circ} + 36 0^{\circ} n, x = 305.26438 \dots^{\circ} + 36 0^{\circ} n, x = 125.26438 \dots^{\circ} + 36 0^{\circ} n, x = - 125.26438 \dots^{\circ} + 36 0^{\circ} n

求解步骤

3 cos (x) = sec (x)

两边减去

sec (x)

3 cos (x) - sec (x) = 0

使用三角恒等式改写

- sec (x) + 3 cos (x)

使用基本三角恒等式:

cos (x) = \frac{1}{sec ( x )}

= - sec (x) + 3 \cdot \frac{1}{sec ( x )}

3 \cdot \frac{1}{sec ( x )} = \frac{3}{sec ( x )}

3 \cdot \frac{1}{sec ( x )}

分式相乘:

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

= \frac{1 \cdot 3}{sec ( x )}

数字相乘:

1 \cdot 3 = 3

= \frac{3}{sec ( x )}

= - sec (x) + \frac{3}{sec ( x )}

\frac{3}{sec ( x )} - sec (x) = 0

用替代法求解

\frac{3}{sec ( x )} - sec (x) = 0

令

: sec (x) = u

\frac{3}{u} - u = 0

\frac{3}{u} - u = 0 : u = 3, u = - 3

\frac{3}{u} - u = 0

在两边乘以

u

\frac{3}{u} - u = 0

在两边乘以

u

\frac{3}{u} u - uu = 0 \cdot u

化简

\frac{3}{u} u - uu = 0 \cdot u

化简

\frac{3}{u} u : 3

\frac{3}{u} u

分式相乘:

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

= \frac{3 u}{u}

约分:

u

= 3

化简

- uu : - u^{2}

- uu

使用指数法则:

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

uu = u^{1 + 1}

= - u^{1 + 1}

数字相加:

1 + 1 = 2

= - u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

3 - u^{2} = 0

3 - u^{2} = 0

3 - u^{2} = 0

解

3 - u^{2} = 0 : u = 3, u = - 3

3 - u^{2} = 0

将

3

到右边

3 - u^{2} = 0

两边减去

3

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

化简

- u^{2} = - 3

- u^{2} = - 3

两边除以

- 1

- u^{2} = - 3

两边除以

- 1

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

化简

u^{2} = 3

u^{2} = 3

对于

x^{2} = f (a)

解为

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

u = 3, u = - 3

u = 3, u = - 3

验证解

找到无定义的点（奇点）:

u = 0

取

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

u = sec (x)

代回

sec (x) = 3, sec (x) = - 3

sec (x) = 3, sec (x) = - 3

sec (x) = 3 : x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

sec (x) = 3

使用反三角函数性质

sec (x) = 3

sec (x) = 3

的通解

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

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

sec (x) = - 3 : x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

sec (x) = - 3

使用反三角函数性质

sec (x) = - 3

sec (x) = - 3

的通解

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

x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

合并所有解

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn, x = arcsec (- 3) + 2 πn, x = - arcsec (- 3) + 2 πn

以小数形式表示解

x = 0.95531 \dots + 2 πn, x = 2 π - 0.95531 \dots + 2 πn, x = 2.18627 \dots + 2 πn, x = - 2.18627 \dots + 2 πn

作图

流行的例子

3csc(x)+5=csc(x)+9

3 csc (x) + 5 = csc (x) + 9

3 = tan^{2} (x)

tan(θ)csc(θ)+2tan(θ)=0

tan (θ) csc (θ) + 2 tan (θ) = 0

4 + 2 csc (\frac{θ}{2}) = 8

2cos(3x)= 1/2 ,x=(17pi)/4

2 cos (3 x) = \frac{1}{2}, x = \frac{17 π}{4}