63. Check My Understanding What are solutions of $$ \sin heta \sin 2 heta=0 $$ on the interval $$ [0,2 \pi) $$ ? Select 4 choice(s) $$ \pi $$ $$ \frac{3 \pi}{2} $$ $$ \frac{3 \pi}{4} $$ $$ \frac{\pi}{3} $$ $$ \frac{\pi}{4} $$ $$ \frac{2 \pi}{3} $$ 0 $$ \frac{\pi}{2} $$

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63. Check My Understanding What are solutions of $$ \sin 	heta \sin 2 	heta=0 $$ on the interval $$ [0,2 \pi) $$ ? Select 4 choice(s) $$ \pi $$ $$ \frac{3 \pi}{2} $$ $$ \frac{3 \pi}{4} $$ $$ \frac{\pi}{3} $$ $$ \frac{\pi}{4} $$ $$ \frac{2 \pi}{3} $$ 0 $$ \frac{\pi}{2} $$

UpStudy AI · Accepted Answer

To solve the equation sin θ · sin 2θ = 0 on the interval [0, 2π), we use the zero product property and set each factor equal to zero:

1. sin θ = 0  
   For θ in [0, 2π), the solutions are:  
   θ = 0 and θ = π

2. sin 2θ = 0  
   Set 2θ = kπ, where k is an integer. This gives:  
   θ = kπ/2  
   For θ in [0, 2π), the valid k values are 0, 1, 2, and 3, corresponding to:  
   θ = 0, θ = π/2, θ = π, and θ = 3π/2

Thus, the distinct solutions in [0, 2π) are:  
0, π/2, π, and 3π/2

Now, based on the provided choices, the correct selections are:  
• π  
• 3π/2  
• 0  
• π/2
The solutions are $$ \pi $$, $$ \frac{3\pi}{2} $$, $$ 0 $$, and $$ \frac{\pi}{2} $$.

63. Check My Understanding What are solutions of \( \sin \theta \sin 2 \theta=0 \) on the interval \( [0,2 \pi) \) ? Select 4 choice(s) \( \pi \) \( \frac{3 \pi}{2} \) \( \frac{3 \pi}{4} \) \( \frac{\pi}{3} \) \( \frac{\pi}{4} \) \( \frac{2 \pi}{3} \) 0 \( \frac{\pi}{2} \)

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