Question

find the general solution of sin∅+ sin3∅ +sin5∅=0 ​

Answer 1

Answer:

θ = 0⁰

Step-by-step explanation:

This question can be done using hit and trial method

Just put 0⁰ everywhere in the place of θ and if it satisfies the condition, then it's true and that value will be the answer...

In this case,

sinθ + sin3θ + sin5θ = 0

If θ = 0⁰,

= sin0⁰ + sin(3×0)⁰ + sin(5×0)⁰

= sin0⁰ + sin0⁰ + sin0⁰

= 0 + 0 + 0

= 0

Hope it helps...

Answer 2

Answer:

Step-by-step explanation:

We have,

$\tt{sin(\theta)+sin(3\theta)+sin(5\theta)=0}$

$\tt{\implies\,sin(3\theta)+2\,sin\left(\dfrac{\theta+5\theta}{2}\right)\,cos\left(\dfrac{\theta-5\theta}{2}\right)=0}$

$\tt{\implies\,sin(3\theta)+2\,sin\left(\dfrac{6\theta}{2}\right)\,cos\left(\dfrac{-4\theta}{2}\right)=0}$

$\tt{\implies\,sin(3\theta)+2\,sin(3\theta)\,cos(2\theta)=0}$

$\tt{\implies\,sin(3\theta)\big\{1+2\,cos(2\theta)\big\}=0}$

$\tt{\implies\,sin(3\theta)=0\,\,\,\,or\,\,\,\,1+2\,cos(2\theta)=0}$

$\tt{\implies\,3\theta=n\pi\,\,\,\,or\,\,\,\,cos(2\theta)=-\dfrac{1}{2}}$

$\tt{\implies\,\theta=\dfrac{n\pi}{3}\,\,\,\,or\,\,\,\,2\theta=2m\pi\pm\dfrac{2\pi}{3}}$

$\tt{\implies\,\theta=\dfrac{n\pi}{3}\,\,\,\,or\,\,\,\,\theta=m\pi\pm\dfrac{\pi}{3}\,\,\,\,\,\,\,\,,\forall\,n,m\in\mathbb{Z}}$