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受欢迎的三角函数 >

5sin(2x)=cos(x)

解答

5 sin (2 x) = cos (x)

解答

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.10016 \dots + 2 πn, x = π - 0.10016 \dots + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 5.73917 \dots^{\circ} + 36 0^{\circ} n, x = 174.26082 \dots^{\circ} + 36 0^{\circ} n

求解步骤

5 sin (2 x) = cos (x)

两边减去

cos (x)

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

使用三角恒等式改写

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

使用倍角公式:

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

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

化简

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

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

分解

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

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

因式分解出通项

cos (x)

= cos (x) (- 1 + 10 sin (x))

cos (x) (10 sin (x) - 1) = 0

分别求解每个部分

cos (x) = 0 or 10 sin (x) - 1 = 0

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

cos (x) = 0

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

10 sin (x) - 1 = 0 : x = arcsin (\frac{1}{10}) + 2 πn, x = π - arcsin (\frac{1}{10}) + 2 πn

10 sin (x) - 1 = 0

将

1

到右边

10 sin (x) - 1 = 0

两边加上

1

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

化简

10 sin (x) = 1

10 sin (x) = 1

两边除以

10

10 sin (x) = 1

两边除以

10

\frac{10 sin ( x )}{10} = \frac{1}{10}

化简

sin (x) = \frac{1}{10}

sin (x) = \frac{1}{10}

使用反三角函数性质

sin (x) = \frac{1}{10}

sin (x) = \frac{1}{10}

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{1}{10}) + 2 πn, x = π - arcsin (\frac{1}{10}) + 2 πn

x = arcsin (\frac{1}{10}) + 2 πn, x = π - arcsin (\frac{1}{10}) + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (\frac{1}{10}) + 2 πn, x = π - arcsin (\frac{1}{10}) + 2 πn

以小数形式表示解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.10016 \dots + 2 πn, x = π - 0.10016 \dots + 2 πn

作图

流行的例子

2sin(x+pi/3)=-1

cot^2(a)=cos^2(a)+cos(a)cos(a)

cot(x)*tan(2x)=3

1*sin(35)=1.33*sin(θ)