solve:- $$ \left(0^{\circ} \leq heta \leq 180^{\circ} ight) $$ a) $$ \sin 4 heta+\sin 2 heta=0 $$

Question

solve:- $$ \left(0^{\circ} \leq 	heta \leq 180^{\circ}ight) $$ a) $$ \sin 4 	heta+\sin 2 	heta=0 $$

UpStudy AI · Accepted Answer

To solve the equation:

$$
\sin 4\theta + \sin 2\theta = 0 \quad \text{for} \quad 0^{\circ} \leq \theta \leq 180^{\circ}
$$

we can use trigonometric identities to simplify and find all possible solutions within the given interval.

### Step 1: Apply Sum-to-Product Identity

Use the identity:

$$
\sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)
$$

Let $$ A = 4\theta $$ and $$ B = 2\theta $$:

$$
\sin 4\theta + \sin 2\theta = 2 \sin\left(\frac{4\theta + 2\theta}{2}\right) \cos\left(\frac{4\theta - 2\theta}{2}\right) = 2 \sin 3\theta \cos \theta
$$

So, the original equation becomes:

$$
2 \sin 3\theta \cos \theta = 0
$$

### Step 2: Set Each Factor to Zero

This equation holds true if either factor is zero:

1. **$$\sin 3\theta = 0$$**
   
   - $$3\theta = 0^{\circ}, 180^{\circ}, 360^{\circ}, 540^{\circ}$$ (since sine is zero at these angles within $$0^{\circ} \leq 3\theta \leq 540^{\circ}$$)
   - Solving for $$\theta$$:
     $$
     \theta = 0^{\circ}, 60^{\circ}, 120^{\circ}, 180^{\circ}
     $$

2. **$$\cos \theta = 0$$**
   
   - $$\cos \theta = 0$$ at $$\theta = 90^{\circ}$$ within $$0^{\circ} \leq \theta \leq 180^{\circ}$$

### Step 3: Combine All Solutions

Collecting all unique solutions from both cases:

$$
\theta = 0^{\circ},\ 60^{\circ},\ 90^{\circ},\ 120^{\circ},\ 180^{\circ}
$$

### Final Answer

All real solutions between 0° and 180° are 0°, 60°, 90°, 120°, and 180°.
The solutions are $$ \theta = 0^\circ, 60^\circ, 90^\circ, 120^\circ, 180^\circ $$.

solve:-
a)

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solve:- a)