Question

solve cos theta + cos 3 theta + cos 5 theta + cos 7 theta =0

Answer 1

Cos theta + cos 3 theta + cos 5theta + cos 7theta =0. Now=2cos4theta.cos3theta + 2cos4theta.costheta=0 Now , cos4theta(2cos2theta.costheta)=0. Now , either cos4theta=0

theta=(2n+1)π/2 and cos2theta=0

theta=(2n+1)π/4 and cos4theta=0

theta=(2n+1)π/8

Answer 2

Answer:

$\theta=(2n+1)\frac{\pi}{2}\cap(2n+1)\frac{\pi}{4}\cap(2n+1)\frac{\pi}{8},n\in \mathbb{Z}$

Step-by-step explanation:

Given : Expression $\cos \theta + \cos 3 \theta + \cos 5 \theta + \cos 7 \theta =0$

To find : Solve the expression ?

Solution :

Applying trigonometric identity,

$\cos a+\cos b=2\cos (\frac{a+b}{2})\cos(\frac{a-b}{2})$

$(\cos \theta + \cos 7 \theta) + (\cos 3 \theta + \cos 5\theta) =0$

$(2\cos(\frac{\theta+7\theta}{2})\cos (\frac{\theta-7\theta}{2}))+ (2\cos (\frac{3\theta+5\theta}{2})\cos (\frac{3\theta-5\theta}{2}) =0$

$2\cos(4\theta)\cos (3\theta)+2\cos (4\theta)\cos (\theta) =0$

$2\cos(4\theta)[\cos (3\theta)+\cos (\theta)] =0$

$2\cos(4\theta)[2\cos(\frac{3\theta+\theta}{2})\cos(\frac{3\theta-\theta}{2})] =0$

$2\cos(4\theta)[2\cos(2\theta)\cos(\theta)] =0$

$4\cos(4\theta)\cos(2\theta)\cos(\theta)=0$

$\cos(\theta)\cos(2\theta)=0$

So,

1) $\cos(\theta)=0$

$\theta=(2n+1)\frac{\pi}{2}$

2) $\cos(2\theta)=0$

$2\theta=(2n+1)\frac{\pi}{2}$

$\theta=(2n+1)\frac{\pi}{4}$

3) $\cos(4\theta)=0$

$4\theta=(2n+1)\frac{\pi}{2}$

$\theta=(2n+1)\frac{\pi}{8}$

The solution of the expression is

$\theta=(2n+1)\frac{\pi}{2}\cap(2n+1)\frac{\pi}{4}\cap(2n+1)\frac{\pi}{8},n\in \mathbb{Z}$