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受欢迎的三角函数 >

20sin(t)cos(t)=-4sin(t)

解答

20 sin (t) cos (t) = - 4 sin (t)

解答

t = 2 πn, t = π + 2 πn, t = 1.77215 \dots + 2 πn, t = - 1.77215 \dots + 2 πn

+1

度数

t = 0^{\circ} + 36 0^{\circ} n, t = 18 0^{\circ} + 36 0^{\circ} n, t = 101.53695 \dots^{\circ} + 36 0^{\circ} n, t = - 101.53695 \dots^{\circ} + 36 0^{\circ} n

求解步骤

20 sin (t) cos (t) = - 4 sin (t)

两边减去

- 4 sin (t)

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

分解

20 sin (t) cos (t) + 4 sin (t) : 4 sin (t) (5 cos (t) + 1)

20 sin (t) cos (t) + 4 sin (t)

将

20

改写为

5 \cdot 4

= 5 \cdot 4 sin (t) cos (t) + 4 sin (t)

因式分解出通项

4 sin (t)

= 4 sin (t) (5 cos (t) + 1)

4 sin (t) (5 cos (t) + 1) = 0

分别求解每个部分

sin (t) = 0 or 5 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

5 cos (t) + 1 = 0 : t = arccos (- \frac{1}{5}) + 2 πn, t = - arccos (- \frac{1}{5}) + 2 πn

5 cos (t) + 1 = 0

将

1

到右边

5 cos (t) + 1 = 0

两边减去

1

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

化简

5 cos (t) = - 1

5 cos (t) = - 1

两边除以

5

5 cos (t) = - 1

两边除以

5

\frac{5 cos ( t )}{5} = \frac{- 1}{5}

化简

cos (t) = - \frac{1}{5}

cos (t) = - \frac{1}{5}

使用反三角函数性质

cos (t) = - \frac{1}{5}

cos (t) = - \frac{1}{5}

的通解

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

t = arccos (- \frac{1}{5}) + 2 πn, t = - arccos (- \frac{1}{5}) + 2 πn

t = arccos (- \frac{1}{5}) + 2 πn, t = - arccos (- \frac{1}{5}) + 2 πn

合并所有解

t = 2 πn, t = π + 2 πn, t = arccos (- \frac{1}{5}) + 2 πn, t = - arccos (- \frac{1}{5}) + 2 πn

以小数形式表示解

t = 2 πn, t = π + 2 πn, t = 1.77215 \dots + 2 πn, t = - 1.77215 \dots + 2 πn

作图

流行的例子

cos(x)-cos(2x)=0,0<= x<360

cos (x) - cos (2 x) = 0, 0^{\circ} \leq x < 36 0^{\circ}

3 = - 3 cos (θ)

2cos(θ)-sqrt(3)=0,0<= θ<= 2pi

2 cos (θ) - 3 = 0, 0 \leq θ \leq 2 π

-4sin(x)=cos^2(x)+1,0<= x<= 2pi

- 4 sin (x) = cos^{2} (x) + 1, 0 \leq x \leq 2 π

tan(2x)cos(2x)+cot(2x)sin(2x)=1

tan (2 x) cos (2 x) + cot (2 x) sin (2 x) = 1