Question

solve the equation sin2thita + sin4thita + sin 6thita =0​

Answer 1

Answer:

principal values for € are 0,π/3,π/2,2π/3,π,4π/3,3π/2,5π/2

Step-by-step explanation:

I am using € instead of theta...

sin 2€ + sin 4€ + sin 6€=0

or,sin 4€ + sin 2€ + sin 6€=0

or,sin 4€ + 2 sin (2€+6€)/2 . cos (2€-6€)/2 =0

or,sin 4€ + 2 sin 4€ . cos 2€=0

or,sin 4€ ( 1 + 2 cos 2€)=0

either sin 4€=0 or (1 + 2 cos 2€)=0

=>4€=2nπ

=>€=nπ/2

n=0 ,€=0

n=1 ,€=π/2

n=2, €=π

n=3, €=3π/2

n=4, €=2π ×××× (as principal value is 0≤€<2π)

or, cos 2€=-1/2

or, 2€=2nπ±2π/3

or,€=nπ±π/3

n=0, €=π/3 (as -π/3<0)

n=1 , €=2π/3 or 4π/3

n=2, €=5π/3 (as 7π/3>2π)

hence, principal values for € are 0,π/3,π/2,2π/3,π,4π/3,3π/2,5π/2....

some information:

*sin A+ sin B= 2 sin (A+B)/2 cos (A-B)/2

*general solutions:

sin a=sin b

=>a=nπ+(-1)ⁿb

cos a=cos b

=>a=2nπ±b

tan a=tan b

=>a=nπ+b

the principal range of value is 0≤€<2π where € is the angle.