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受欢迎的三角函数 >

4cos^2(x)+4sqrt(3)cos(x)+3=0

解答

4 cos^{2} (x) + 43 cos (x) + 3 = 0

解答

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

+1

度数

x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

求解步骤

4 cos^{2} (x) + 43 cos (x) + 3 = 0

用替代法求解

4 cos^{2} (x) + 43 cos (x) + 3 = 0

令

: cos (x) = u

4 u^{2} + 43 u + 3 = 0

4 u^{2} + 43 u + 3 = 0 : u = - \frac{3}{2}

4 u^{2} + 43 u + 3 = 0

使用求根公式求解

4 u^{2} + 43 u + 3 = 0

二次方程求根公式:

若

a = 4, b = 43, c = 3

u_{1, 2} = \frac{- 4 3 \pm ( 4 3 ) ^{2} - 4 \cdot 4 \cdot 3}{2 \cdot 4}

u_{1, 2} = \frac{- 4 3 \pm ( 4 3 ) ^{2} - 4 \cdot 4 \cdot 3}{2 \cdot 4}

(43)^{2} - 4 \cdot 4 \cdot 3 = 0

(43)^{2} - 4 \cdot 4 \cdot 3

(43)^{2} = 4^{2} \cdot 3

(43)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 4^{2} (3)^{2}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 3

= 4^{2} \cdot 3

4 \cdot 4 \cdot 3 = 48

4 \cdot 4 \cdot 3

数字相乘:

4 \cdot 4 \cdot 3 = 48

= 48

= 4^{2} \cdot 3 - 48

4^{2} \cdot 3 = 48

4^{2} \cdot 3

4^{2} = 16

= 16 \cdot 3

数字相乘:

16 \cdot 3 = 48

= 48

= 48 - 48

数字相减:

48 - 48 = 0

= 0

u_{1, 2} = \frac{- 4 3 \pm 0}{2 \cdot 4}

u = \frac{- 4 3}{2 \cdot 4}

\frac{- 4 3}{2 \cdot 4} = - \frac{3}{2}

\frac{- 4 3}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 4 3}{8}

使用分式法则:

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

= - \frac{4 3}{8}

约分:

4

= - \frac{3}{2}

u = - \frac{3}{2}

二次方程组的解是:

u = - \frac{3}{2}

u = cos (x)

代回

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2}

的通解

cos (x)

周期表（周期为

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

合并所有解

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

作图

流行的例子

4cos^2(β)-4cos(β)=3

4 cos^{2} (β) - 4 cos (β) = 3

solvefor x,cot(x)=1

so l v e f or x, cot (x) = 1

sin^4(x)+cos^4(x)= 5/8

sin^{4} (x) + cos^{4} (x) = \frac{5}{8}

0 = \frac{1}{9} sin (9 t)

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