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受欢迎的三角函数 >

4sin(x)cos(x)-2sqrt(3)*sin(x)-2cos(x)+sqrt(3)=0

解答

4 sin (x) cos (x) - 23 \cdot sin (x) - 2 cos (x) + 3 = 0

解答

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

4 sin (x) cos (x) - 23 sin (x) - 2 cos (x) + 3 = 0

分解

4 sin (x) cos (x) - 23 sin (x) - 2 cos (x) + 3 : (- 1 + 2 sin (x)) (2 cos (x) - 3)

4 sin (x) cos (x) - 23 sin (x) - 2 cos (x) + 3

分解

4 sin (x) cos (x) - 2 cos (x) : 2 cos (x) (2 sin (x) - 1)

4 sin (x) cos (x) - 2 cos (x)

改写为

= 2 \cdot 2 cos (x) sin (x) - 1 \cdot 2 cos (x)

因式分解出通项

2 cos (x)

= 2 cos (x) (2 sin (x) - 1)

分解

- 23 sin (x) + 3 : 3 (- 2 sin (x) + 1)

- 23 sin (x) + 3

因式分解出通项

3

= 3 (- 2 sin (x) + 1)

= 2 cos (x) (2 sin (x) - 1) + 3 (- 2 sin (x) + 1)

改写为

= 2 (- 1 + 2 sin (x)) cos (x) - (- 1 + 2 sin (x)) 3

因式分解出通项

(- 1 + 2 sin (x))

= (- 1 + 2 sin (x)) (2 cos (x) - 3)

(- 1 + 2 sin (x)) (2 cos (x) - 3) = 0

分别求解每个部分

- 1 + 2 sin (x) = 0 or 2 cos (x) - 3 = 0

- 1 + 2 sin (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

- 1 + 2 sin (x) = 0

将

1

到右边

- 1 + 2 sin (x) = 0

两边加上

1

- 1 + 2 sin (x) + 1 = 0 + 1

化简

2 sin (x) = 1

2 sin (x) = 1

两边除以

2

2 sin (x) = 1

两边除以

2

\frac{2 sin ( x )}{2} = \frac{1}{2}

化简

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

的通解

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

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

2 cos (x) - 3 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

2 cos (x) - 3 = 0

将

3

到右边

2 cos (x) - 3 = 0

两边加上

3

2 cos (x) - 3 + 3 = 0 + 3

化简

2 cos (x) = 3

2 cos (x) = 3

两边除以

2

2 cos (x) = 3

两边除以

2

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

化简

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

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{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

作图

流行的例子

4sin^2(x)-7sin(x)+3=0

4 sin^{2} (x) - 7 sin (x) + 3 = 0

2sin(x-10)=-0.4

2 sin (x - 1 0^{\circ}) = - 0.4

cot^2(2x-pi/3)= 1/3

cot^{2} (2 x - \frac{π}{3}) = \frac{1}{3}

cos (2 π x) = 0

sin(2x)=0.8,-180<= x<= 180

sin (2 x) = 0.8, - 18 0^{\circ} \leq x \leq 18 0^{\circ}