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受欢迎的三角函数 >

5sin(2x)=5cos(x),0<= x<= 2pi

解答

5 sin (2 x) = 5 cos (x), 0 \leq x \leq 2 π

解答

x = \frac{π}{2}, x = \frac{3 π}{2}, x = \frac{π}{6}, x = \frac{5 π}{6}

+1

度数

x = 9 0^{\circ}, x = 27 0^{\circ}, x = 3 0^{\circ}, x = 15 0^{\circ}

求解步骤

5 sin (2 x) = 5 cos (x), 0 \leq x \leq 2 π

两边减去

5 cos (x)

5 sin (2 x) - 5 cos (x) = 0

使用三角恒等式改写

- 5 cos (x) + 5 sin (2 x)

使用倍角公式:

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

= - 5 cos (x) + 5 \cdot 2 sin (x) cos (x)

化简

= - 5 cos (x) + 10 sin (x) cos (x)

- 5 cos (x) + 10 cos (x) sin (x) = 0

分解

- 5 cos (x) + 10 cos (x) sin (x) : 5 cos (x) (2 sin (x) - 1)

- 5 cos (x) + 10 cos (x) sin (x)

将

10

改写为

2 \cdot 5

= - 5 cos (x) + 2 \cdot 5 sin (x) cos (x)

因式分解出通项

5 cos (x)

= 5 cos (x) (- 1 + 2 sin (x))

5 cos (x) (2 sin (x) - 1) = 0

分别求解每个部分

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

cos (x) = 0, 0 \leq x \leq 2 π : x = \frac{π}{2}, x = \frac{3 π}{2}

cos (x) = 0, 0 \leq x \leq 2 π

cos (x) = 0

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}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

在

0 \leq x \leq 2 π

范围内的解

x = \frac{π}{2}, x = \frac{3 π}{2}

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

2 sin (x) - 1 = 0, 0 \leq x \leq 2 π

将

1

到右边

2 sin (x) - 1 = 0

两边加上

1

2 sin (x) - 1 + 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

在

0 \leq x \leq 2 π

范围内的解

x = \frac{π}{6}, x = \frac{5 π}{6}

合并所有解

x = \frac{π}{2}, x = \frac{3 π}{2}, x = \frac{π}{6}, x = \frac{5 π}{6}

作图

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