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受欢迎的三角函数 >

6sin(2t)+9sin(t)=0

解答

6 sin (2 t) + 9 sin (t) = 0

解答

t = 2 πn, t = π + 2 πn, t = 2.41885 \dots + 2 πn, t = - 2.41885 \dots + 2 πn

+1

度数

t = 0^{\circ} + 36 0^{\circ} n, t = 18 0^{\circ} + 36 0^{\circ} n, t = 138.59037 \dots^{\circ} + 36 0^{\circ} n, t = - 138.59037 \dots^{\circ} + 36 0^{\circ} n

求解步骤

6 sin (2 t) + 9 sin (t) = 0

使用三角恒等式改写

6 sin (2 t) + 9 sin (t)

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

= 6 \cdot 2 sin (t) cos (t) + 9 sin (t)

化简

= 12 sin (t) cos (t) + 9 sin (t)

9 sin (t) + 12 cos (t) sin (t) = 0

分解

9 sin (t) + 12 cos (t) sin (t) : 3 sin (t) (4 cos (t) + 3)

9 sin (t) + 12 cos (t) sin (t)

将

12

改写为

4 \cdot 3

将

9

改写为

3 \cdot 3

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

因式分解出通项

3 sin (t)

= 3 sin (t) (3 + 4 cos (t))

3 sin (t) (4 cos (t) + 3) = 0

分别求解每个部分

sin (t) = 0 or 4 cos (t) + 3 = 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

4 cos (t) + 3 = 0 : t = arccos (- \frac{3}{4}) + 2 πn, t = - arccos (- \frac{3}{4}) + 2 πn

4 cos (t) + 3 = 0

将

3

到右边

4 cos (t) + 3 = 0

两边减去

3

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

化简

4 cos (t) = - 3

4 cos (t) = - 3

两边除以

4

4 cos (t) = - 3

两边除以

4

\frac{4 cos ( t )}{4} = \frac{- 3}{4}

化简

cos (t) = - \frac{3}{4}

cos (t) = - \frac{3}{4}

使用反三角函数性质

cos (t) = - \frac{3}{4}

cos (t) = - \frac{3}{4}

的通解

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

t = arccos (- \frac{3}{4}) + 2 πn, t = - arccos (- \frac{3}{4}) + 2 πn

t = arccos (- \frac{3}{4}) + 2 πn, t = - arccos (- \frac{3}{4}) + 2 πn

合并所有解

t = 2 πn, t = π + 2 πn, t = arccos (- \frac{3}{4}) + 2 πn, t = - arccos (- \frac{3}{4}) + 2 πn

以小数形式表示解

t = 2 πn, t = π + 2 πn, t = 2.41885 \dots + 2 πn, t = - 2.41885 \dots + 2 πn

作图

流行的例子

sin(2x+60)+sin(x+30)=0

sin (2 x + 60) + sin (x + 30) = 0

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

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

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

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

4 sec (x) + 8 = 0

solvefor x,ln(y)=e^ysin(x)

so l v e f or x, ln (y) = e^{y} sin (x)