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受欢迎的三角函数 >

cos(t)=sin(2t)

解答

cos (t) = sin (2 t)

解答

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn, t = \frac{π}{6} + 2 πn, t = \frac{5 π}{6} + 2 πn

+1

度数

t = 9 0^{\circ} + 36 0^{\circ} n, t = 27 0^{\circ} + 36 0^{\circ} n, t = 3 0^{\circ} + 36 0^{\circ} n, t = 15 0^{\circ} + 36 0^{\circ} n

求解步骤

cos (t) = sin (2 t)

两边减去

sin (2 t)

cos (t) - sin (2 t) = 0

使用三角恒等式改写

cos (t) - sin (2 t)

使用倍角公式:

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

= cos (t) - 2 sin (t) cos (t)

cos (t) - 2 cos (t) sin (t) = 0

分解

cos (t) - 2 cos (t) sin (t) : - cos (t) (2 sin (t) - 1)

cos (t) - 2 cos (t) sin (t)

因式分解出通项

- cos (t)

= - cos (t) (- 1 + 2 sin (t))

- cos (t) (2 sin (t) - 1) = 0

分别求解每个部分

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

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

cos (t) = 0

cos (t) = 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}

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

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

2 sin (t) - 1 = 0 : t = \frac{π}{6} + 2 πn, t = \frac{5 π}{6} + 2 πn

2 sin (t) - 1 = 0

将

1

到右边

2 sin (t) - 1 = 0

两边加上

1

2 sin (t) - 1 + 1 = 0 + 1

化简

2 sin (t) = 1

2 sin (t) = 1

两边除以

2

2 sin (t) = 1

两边除以

2

\frac{2 sin ( t )}{2} = \frac{1}{2}

化简

sin (t) = \frac{1}{2}

sin (t) = \frac{1}{2}

sin (t) = \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}

t = \frac{π}{6} + 2 πn, t = \frac{5 π}{6} + 2 πn

t = \frac{π}{6} + 2 πn, t = \frac{5 π}{6} + 2 πn

合并所有解

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn, t = \frac{π}{6} + 2 πn, t = \frac{5 π}{6} + 2 πn

作图

流行的例子

cos^2(111)+cos^2(69.3)+cos^2(x)=1

cos^{2} (11 1^{\circ}) + cos^{2} (69. 3^{\circ}) + cos^{2} (x) = 1

4+2cos(2x)=6cos(x)

4 + 2 cos (2 x) = 6 cos (x)

cos(x)-tan(x)cos(x)=0

cos (x) - tan (x) cos (x) = 0

cot (2 θ) = \frac{3}{4}

tan(x)=-4/(sqrt(33))

tan (x) = - \frac{4}{33}