If $cos~ heta+cos~2 heta+cos~3 heta=0,$ then general solution is equal to :

Q: If $cos~\theta+cos~2\theta+cos~3\theta=0,$ then general solution is equal to :

If $cos~\theta+cos~2\theta+cos~3\theta=0,$ then general solution is equal to :

Question

If $cos~	heta+cos~2	heta+cos~3	heta=0,$ then general solution is equal to :

Accepted Answer

To solve the problem where $ \cos \theta + \cos 2\theta + \cos 3\theta = 0 $, we need to find the general solution for $\theta$.

### Explanation for Option 1:

The expression $\cos \theta + \cos 2\theta + \cos 3\theta = 0$ can be solved using trigonometric identities and properties. We can use the sum-to-product identities to simplify the expression:

1. **Sum-to-Product Identity**: 
   $$
   \cos A + \cos B = 2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)
   $$

Applying this identity to $\cos \theta + \cos 3\theta$:
   $$
   \cos \theta + \cos 3\theta = 2 \cos \left(\frac{4\theta}{2}\right) \cos \left(\frac{-2\theta}{2}\right) = 2 \cos 2\theta \cos (-\theta) = 2 \cos 2\theta \cos \theta
   $$

So, the equation becomes:
   $$
   2 \cos 2\theta \cos \theta + \cos 2\theta = 0
   $$

2. **Factor the Expression**:
   $$
   \cos 2\theta (2 \cos \theta + 1) = 0
   $$

This gives us two cases to solve:
   - $\cos 2\theta = 0$
   - $2 \cos \theta + 1 = 0$

3. **Solving $\cos 2\theta = 0$**:
   $$
   2\theta = (2n+1)\frac{\pi}{2} \quad \Rightarrow \quad \theta = \frac{(2n+1)\pi}{4}
   $$

4. **Solving $2 \cos \theta + 1 = 0$**:
   $$
   \cos \theta = -\frac{1}{2}
   $$
   The general solution for $\cos \theta = -\frac{1}{2}$ is:
   $$
   \theta = 2n\pi \pm \frac{2\pi}{3}
   $$

Thus, the general solution for the equation $\cos \theta + \cos 2\theta + \cos 3\theta = 0$ is $\theta = 2n\pi \pm \frac{2\pi}{3}$, which corresponds to **Option 1**.

### Explanation for Why Other Options are Incorrect:

- **Option 2: $n\pi \pm \frac{\pi}{3}$**:
  This solution does not satisfy the equation $\cos \theta + \cos 2\theta + \cos 3\theta = 0$ for all $n$. The angles derived from this option do not consistently result in the sum being zero.

- **Option 3: $2n\pi \pm \frac{\pi}{3}$**:
  While this might seem similar to the correct solution, it does not account for the correct periodicity and symmetry required by the equation. The factor of $\frac{\pi}{3}$ alone does not satisfy the equation for all $n$.

- **Option 4: $n\pi \pm \frac{2\pi}{3}$**:
  This option suggests a different periodicity that does not align with the derived solutions from the trigonometric identities used. The factor of $n\pi$ does not correctly capture the behavior of the cosine terms in the original equation.

In conclusion, **Option 1** is the correct answer because it accurately reflects the general solution derived from the trigonometric identities and properties applied to the given equation.