What is the general solution for sin(x)+sin(5x)=0 ?

The general solution for sin(x)+sin(5x)=0 is x= pi/4+pin,x=(3pi)/4+pin,x=(2pin)/3 ,x= pi/3+(2pin)/3

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sin(x)+sin(5x)=0

sin (x) + sin (5 x) = 0

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn, x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

Solution steps

sin (x) + sin (5 x) = 0

Rewrite using trig identities

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

Solving each part separately

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

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

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

Combine all the solutions

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn, x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}

sin^{2} (x) + cos^{2} (x) = tan (x)

3 tan (3 x) = tan (x)

sin (x) = \frac{5}{5}, sin (2 x)

0 = 1 - cos (2 π x)

tan (θ) = \frac{12}{5}, 0 \leq θ \leq \frac{π}{2}

What is the general solution for sin(x)+sin(5x)=0 ?
The general solution for sin(x)+sin(5x)=0 is x= pi/4+pin,x=(3pi)/4+pin,x=(2pin)/3 ,x= pi/3+(2pin)/3