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受欢迎的三角函数 >

12sin(t)cos(t)=8sin(t)

解答

12 sin (t) cos (t) = 8 sin (t)

解答

t = 2 πn, t = π + 2 πn, t = 0.84106 \dots + 2 πn, t = 2 π - 0.84106 \dots + 2 πn

+1

度数

t = 0^{\circ} + 36 0^{\circ} n, t = 18 0^{\circ} + 36 0^{\circ} n, t = 48.18968 \dots^{\circ} + 36 0^{\circ} n, t = 311.81031 \dots^{\circ} + 36 0^{\circ} n

求解步骤

12 sin (t) cos (t) = 8 sin (t)

两边减去

8 sin (t)

12 sin (t) cos (t) - 8 sin (t) = 0

分解

12 sin (t) cos (t) - 8 sin (t) : 4 sin (t) (3 cos (t) - 2)

12 sin (t) cos (t) - 8 sin (t)

将

- 8

改写为

2 \cdot 4

将

12

改写为

3 \cdot 4

= 3 \cdot 4 sin (t) cos (t) + 2 \cdot 4 sin (t)

因式分解出通项

4 sin (t)

= 4 sin (t) (3 cos (t) - 2)

4 sin (t) (3 cos (t) - 2) = 0

分别求解每个部分

sin (t) = 0 or 3 cos (t) - 2 = 0

sin (t) = 0 : t = 2 πn, t = π + 2 πn

sin (t) = 0

sin (t) = 0

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}

t = 0 + 2 πn, t = π + 2 πn

t = 0 + 2 πn, t = π + 2 πn

解

t = 0 + 2 πn : t = 2 πn

t = 0 + 2 πn

0 + 2 πn = 2 πn

t = 2 πn

t = 2 πn, t = π + 2 πn

3 cos (t) - 2 = 0 : t = arccos (\frac{2}{3}) + 2 πn, t = 2 π - arccos (\frac{2}{3}) + 2 πn

3 cos (t) - 2 = 0

将

2

到右边

3 cos (t) - 2 = 0

两边加上

2

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

化简

3 cos (t) = 2

3 cos (t) = 2

两边除以

3

3 cos (t) = 2

两边除以

3

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

化简

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

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

使用反三角函数性质

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

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

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

t = arccos (\frac{2}{3}) + 2 πn, t = 2 π - arccos (\frac{2}{3}) + 2 πn

t = arccos (\frac{2}{3}) + 2 πn, t = 2 π - arccos (\frac{2}{3}) + 2 πn

合并所有解

t = 2 πn, t = π + 2 πn, t = arccos (\frac{2}{3}) + 2 πn, t = 2 π - arccos (\frac{2}{3}) + 2 πn

以小数形式表示解

t = 2 πn, t = π + 2 πn, t = 0.84106 \dots + 2 πn, t = 2 π - 0.84106 \dots + 2 πn

作图

流行的例子

4sin^2(x)-8sin(x)+3=0

4 sin^{2} (x) - 8 sin (x) + 3 = 0

3tan(2x)+sqrt(3)=0

3 tan (2 x) + 3 = 0

tan(2x+13)=cot(x+53)

tan (2 x + 13) = cot (x + 5 3^{\circ})

sin(2x)-sqrt(3)sin(x)=0

sin (2 x) - 3 sin (x) = 0

2sin(2x)+sqrt(3)cos(x)=0

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