What is the general solution for sin(x)+sin(3x)=0 ?

The general solution for sin(x)+sin(3x)=0 is x=pin,x= pi/2+pin

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

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

x = πn, x = \frac{π}{2} + πn

Solution steps

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

Rewrite using trig identities

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

Solving each part separately

cos (x) = 0 or sin (2 x) = 0

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

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

Combine all the solutions

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

Merge Overlapping Intervals

x = πn, x = \frac{π}{2} + πn

2 sin^{2} (θ) - 5 sin (θ) + 3 = 0

2 sin (x) csc (x) - csc (x) = 4 sin (x) - 2

2 cos^{2} (x) + 5 cos (x) + 2 = 0

sin (θ) = - 0.66

4 sec^{2} (θ) tan (θ) = 8 tan (θ)

What is the general solution for sin(x)+sin(3x)=0 ?
The general solution for sin(x)+sin(3x)=0 is x=pin,x= pi/2+pin