What is the general solution for sin(2A)=sin(A) ?

The general solution for sin(2A)=sin(A) is A=2pin,A=pi+2pin,A= pi/3+2pin,A=(5pi)/3+2pin

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sin(2A)=sin(A)

sin (2 A) = sin (A)

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

Solution steps

sin (2 A) = sin (A)

Subtract

sin (A)

from both sides

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

sin (A) = 0 : 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

Combine all the solutions

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

csc (θ) = \frac{3}{2}

\frac{cos ( θ )}{sin ( θ )} = - 1

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

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

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

What is the general solution for sin(2A)=sin(A) ?
The general solution for sin(2A)=sin(A) is A=2pin,A=pi+2pin,A= pi/3+2pin,A=(5pi)/3+2pin