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受欢迎的三角函数 >

5cot^2(x)+8cot(x)+3=0

解答

5 cot^{2} (x) + 8 cot (x) + 3 = 0

解答

x = 2.11121 \dots + πn, x = \frac{3 π}{4} + πn

+1

度数

x = 120.96375 \dots^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

求解步骤

5 cot^{2} (x) + 8 cot (x) + 3 = 0

用替代法求解

5 cot^{2} (x) + 8 cot (x) + 3 = 0

令

: cot (x) = u

5 u^{2} + 8 u + 3 = 0

5 u^{2} + 8 u + 3 = 0 : u = - \frac{3}{5}, u = - 1

5 u^{2} + 8 u + 3 = 0

使用求根公式求解

5 u^{2} + 8 u + 3 = 0

二次方程求根公式:

若

a = 5, b = 8, c = 3

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 5 \cdot 3}{2 \cdot 5}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 \cdot 5 \cdot 3}{2 \cdot 5}

8^{2} - 4 \cdot 5 \cdot 3 = 2

8^{2} - 4 \cdot 5 \cdot 3

数字相乘:

4 \cdot 5 \cdot 3 = 60

= 8^{2} - 60

8^{2} = 64

= 64 - 60

数字相减:

64 - 60 = 4

= 4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

u_{1, 2} = \frac{- 8 \pm 2}{2 \cdot 5}

将解分隔开

u_{1} = \frac{- 8 + 2}{2 \cdot 5}, u_{2} = \frac{- 8 - 2}{2 \cdot 5}

u = \frac{- 8 + 2}{2 \cdot 5} : - \frac{3}{5}

\frac{- 8 + 2}{2 \cdot 5}

数字相加/相减:

- 8 + 2 = - 6

= \frac{- 6}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{- 6}{10}

使用分式法则:

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

= - \frac{6}{10}

约分:

2

= - \frac{3}{5}

u = \frac{- 8 - 2}{2 \cdot 5} : - 1

\frac{- 8 - 2}{2 \cdot 5}

数字相减:

- 8 - 2 = - 10

= \frac{- 10}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{- 10}{10}

使用分式法则:

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

= - \frac{10}{10}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

u = - \frac{3}{5}, u = - 1

u = cot (x)

代回

cot (x) = - \frac{3}{5}, cot (x) = - 1

cot (x) = - \frac{3}{5}, cot (x) = - 1

cot (x) = - \frac{3}{5} : x = arccot (- \frac{3}{5}) + πn

cot (x) = - \frac{3}{5}

使用反三角函数性质

cot (x) = - \frac{3}{5}

cot (x) = - \frac{3}{5}

的通解

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- \frac{3}{5}) + πn

x = arccot (- \frac{3}{5}) + πn

cot (x) = - 1 : x = \frac{3 π}{4} + πn

cot (x) = - 1

cot (x) = - 1

的通解

cot (x)

周期表（周期为

πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

合并所有解

x = arccot (- \frac{3}{5}) + πn, x = \frac{3 π}{4} + πn

以小数形式表示解

x = 2.11121 \dots + πn, x = \frac{3 π}{4} + πn

作图

流行的例子

sin (x) = \frac{15}{31}

4 cot (θ) - 1 = 0

solvefor x,4cos^2(x)=5-4sin(x)

so l v e f or x, 4 cos^{2} (x) = 5 - 4 sin (x)

|cos(x)|=(sqrt(3))/2

∣ cos (x) ∣ = \frac{3}{2}

tan(θ)=(-1)/(sqrt(3)),0<= θ<= 360

tan (θ) = \frac{- 1}{3}, 0^{\circ} \leq θ \leq 36 0^{\circ}