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受欢迎的三角函数 >

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

解答

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

解答

x = \frac{π}{2}, x = \frac{3 π}{2}

+1

度数

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

求解步骤

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

使用三角恒等式改写

sin (2 x) + 2 cos (x)

使用倍角公式:

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

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

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

分解

2 cos (x) + 2 cos (x) sin (x) : 2 cos (x) (sin (x) + 1)

2 cos (x) + 2 cos (x) sin (x)

因式分解出通项

2 cos (x)

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

2 cos (x) (sin (x) + 1) = 0

分别求解每个部分

cos (x) = 0 or 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}

sin (x) + 1 = 0, 0 \leq x \leq 2 π : x = \frac{3 π}{2}

sin (x) + 1 = 0, 0 \leq x \leq 2 π

将

1

到右边

sin (x) + 1 = 0

两边减去

1

sin (x) + 1 - 1 = 0 - 1

化简

sin (x) = - 1

sin (x) = - 1

sin (x) = - 1

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{3 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{3 π}{2}

合并所有解

x = \frac{π}{2}, x = \frac{3 π}{2}

作图

流行的例子

tan^2(x)sin(x)=sin(x)

tan^{2} (x) sin (x) = sin (x)

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

2 sin (2 x) - 3 = 0

2sin(x)cos(x)=0

2 sin (x) cos (x) = 0

tan^5(x)-9tan(x)=0

tan^{5} (x) - 9 tan (x) = 0

1 - cos (θ) = \frac{1}{2}