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受欢迎的三角函数 >

18sin(t)cos(t)=6sin(t)

解答

18 sin (t) cos (t) = 6 sin (t)

解答

t = 2 πn, t = π + 2 πn, t = 1.23095 \dots + 2 πn, t = 2 π - 1.23095 \dots + 2 πn

+1

度数

t = 0^{\circ} + 36 0^{\circ} n, t = 18 0^{\circ} + 36 0^{\circ} n, t = 70.52877 \dots^{\circ} + 36 0^{\circ} n, t = 289.47122 \dots^{\circ} + 36 0^{\circ} n

求解步骤

18 sin (t) cos (t) = 6 sin (t)

两边减去

6 sin (t)

18 sin (t) cos (t) - 6 sin (t) = 0

分解

18 sin (t) cos (t) - 6 sin (t) : 6 sin (t) (3 cos (t) - 1)

18 sin (t) cos (t) - 6 sin (t)

将

18

改写为

3 \cdot 6

= 3 \cdot 6 sin (t) cos (t) - 6 sin (t)

因式分解出通项

6 sin (t)

= 6 sin (t) (3 cos (t) - 1)

6 sin (t) (3 cos (t) - 1) = 0

分别求解每个部分

sin (t) = 0 or 3 cos (t) - 1 = 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) - 1 = 0 : t = arccos (\frac{1}{3}) + 2 πn, t = 2 π - arccos (\frac{1}{3}) + 2 πn

3 cos (t) - 1 = 0

将

1

到右边

3 cos (t) - 1 = 0

两边加上

1

3 cos (t) - 1 + 1 = 0 + 1

化简

3 cos (t) = 1

3 cos (t) = 1

两边除以

3

3 cos (t) = 1

两边除以

3

\frac{3 cos ( t )}{3} = \frac{1}{3}

化简

cos (t) = \frac{1}{3}

cos (t) = \frac{1}{3}

使用反三角函数性质

cos (t) = \frac{1}{3}

cos (t) = \frac{1}{3}

的通解

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

t = arccos (\frac{1}{3}) + 2 πn, t = 2 π - arccos (\frac{1}{3}) + 2 πn

t = arccos (\frac{1}{3}) + 2 πn, t = 2 π - arccos (\frac{1}{3}) + 2 πn

合并所有解

t = 2 πn, t = π + 2 πn, t = arccos (\frac{1}{3}) + 2 πn, t = 2 π - arccos (\frac{1}{3}) + 2 πn

以小数形式表示解

t = 2 πn, t = π + 2 πn, t = 1.23095 \dots + 2 πn, t = 2 π - 1.23095 \dots + 2 πn

作图

流行的例子

8sin(x)=sqrt(41-40cos(x))

8 sin (x) = 41 - 40 cos (x)

2sin^2(x)+9sin(x)+4=0

2 sin^{2} (x) + 9 sin (x) + 4 = 0

6sin(2x)=6cos(x)

6 sin (2 x) = 6 cos (x)

cot (θ) = \frac{5}{12}

tan(x)=(7.4)/(9.3)

tan (x) = \frac{7.4}{9.3}