Question

solve this cos 7 theta =
cos theta minus sin 4 theta​

Answer 1

Given: $cos7\theta=cos\theta-sin4\theta$

To find: the solution

Solution:

Now, $cos7\theta=cos\theta-sin4\theta$

$\Rightarrow cos7\theta-cos\theta+sin4\theta=0$

$\Rightarrow 2\:sin\frac{7\theta+\theta}{2}\:sin\frac{\theta-7\theta}{2}+sin4\theta=0$

$\Rightarrow -2\:sin4\theta\:sin3\theta+sin4\theta=0$

$\Rightarrow sin4\theta\:(2\:sin3\theta-1)$

$\Rightarrow sin4\theta=0\:or,\:2\:sin3\theta=1$

$\Rightarrow sin4\theta=sin0\:or,\:sin3\theta=\frac{1}{2}=sin\frac{\pi}{6}$

$\Rightarrow 4\theta=0\:or,\:3\theta=\frac{\pi}{6}$

$\Rightarrow \theta=0\:or,\:\theta=\frac{\pi}{18}$

Answer: required solution is $\theta=0,\:\frac{\pi}{18}$

Remember:

• $cosC+cosD=2\:sin\frac{C+D}{2}\:sin\frac{C-D}{2}$

• $cosC-cosD=2\:sin\frac{C+D}{2}\:sin\frac{D-C}{2}$

Answer 2

Step-by-step explanation:

Cos 7 theta - Cos theta + Sin theta =0

2Sin 8 theta/2 Sin 6 theta/2 + Sin 4 theta =0

2Sin 4 theta Sin 3 theta + Sin 4 theta =0

Sin 4 theta (2Sin 3 theta + 1)=0

Sin 4 theta =0 OR 2 Sin 3 theta +1 =0

4 theta =npi OR Sin 3 theta = -1/2