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受欢迎的三角函数 >

sin(2A)=sin(A)

解答

sin (2 A) = sin (A)

解答

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

+1

度数

A = 0^{\circ} + 36 0^{\circ} n, A = 18 0^{\circ} + 36 0^{\circ} n, A = 6 0^{\circ} + 36 0^{\circ} n, A = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

sin (2 A) = sin (A)

两边减去

sin (A)

sin (2 A) - sin (A) = 0

使用三角恒等式改写

sin (2 A) - sin (A)

使用倍角公式:

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

= 2 sin (A) cos (A) - sin (A)

- sin (A) + 2 cos (A) sin (A) = 0

分解

- sin (A) + 2 cos (A) sin (A) : sin (A) (2 cos (A) - 1)

- sin (A) + 2 cos (A) sin (A)

因式分解出通项

sin (A)

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

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

分别求解每个部分

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

sin (A) = 0 : A = 2 πn, A = π + 2 πn

sin (A) = 0

sin (A) = 0

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}

A = 0 + 2 πn, A = π + 2 πn

A = 0 + 2 πn, A = π + 2 πn

解

A = 0 + 2 πn : A = 2 πn

A = 0 + 2 πn

0 + 2 πn = 2 πn

A = 2 πn

A = 2 πn, A = π + 2 πn

2 cos (A) - 1 = 0 : A = \frac{π}{3} + 2 πn, A = \frac{5 π}{3} + 2 πn

2 cos (A) - 1 = 0

将

1

到右边

2 cos (A) - 1 = 0

两边加上

1

2 cos (A) - 1 + 1 = 0 + 1

化简

2 cos (A) = 1

2 cos (A) = 1

两边除以

2

2 cos (A) = 1

两边除以

2

\frac{2 cos ( A )}{2} = \frac{1}{2}

化简

cos (A) = \frac{1}{2}

cos (A) = \frac{1}{2}

cos (A) = \frac{1}{2}

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}

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

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

合并所有解

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

作图

流行的例子

csc (θ) = \frac{3}{2}

(cos(θ))/(sin(θ))=-1

\frac{cos ( θ )}{sin ( θ )} = - 1

cos (B) = \frac{12}{13}

sin(θ)=sqrt(2/9),θ,3r(θ,d)d,tan(2θ)

sin (θ) = \frac{2}{9}, θ, 3 r (θ, d) d, tan (2 θ)

sin(x/2)=0,0<= x<= 2pi

sin (\frac{x}{2}) = 0, 0 \leq x \leq 2 π