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受欢迎的三角函数 >

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

解答

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

解答

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 131.40962 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

- 2 sin (2 x) + 3 cos (x)

使用倍角公式:

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

= - 2 \cdot 2 sin (x) cos (x) + 3 cos (x)

化简

= - 4 sin (x) cos (x) + 3 cos (x)

3 cos (x) - 4 cos (x) sin (x) = 0

分解

3 cos (x) - 4 cos (x) sin (x) : - cos (x) (4 sin (x) - 3)

3 cos (x) - 4 cos (x) sin (x)

因式分解出通项

- cos (x)

= - cos (x) (- 3 + 4 sin (x))

- cos (x) (4 sin (x) - 3) = 0

分别求解每个部分

cos (x) = 0 or 4 sin (x) - 3 = 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

4 sin (x) - 3 = 0 : x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

4 sin (x) - 3 = 0

将

3

到右边

4 sin (x) - 3 = 0

两边加上

3

4 sin (x) - 3 + 3 = 0 + 3

化简

4 sin (x) = 3

4 sin (x) = 3

两边除以

4

4 sin (x) = 3

两边除以

4

\frac{4 sin ( x )}{4} = \frac{3}{4}

化简

sin (x) = \frac{3}{4}

sin (x) = \frac{3}{4}

使用反三角函数性质

sin (x) = \frac{3}{4}

sin (x) = \frac{3}{4}

的通解

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

x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

以小数形式表示解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

作图

流行的例子

4 cos (2 t) = 2

tan(θ)=(10)/(5.32)

tan (θ) = \frac{10}{5.32}

2 cos (x) = \frac{6}{7}

tan(θ)= 1/2 ,pi<θ<(3pi)/2

tan (θ) = \frac{1}{2}, π < θ < \frac{3 π}{2}

7/2 =3sin(b(pi/2))+2

\frac{7}{2} = 3 sin (b (\frac{π}{2})) + 2