Question

The equation $\sin \frac{\theta}{2}\, (\sin \frac{\theta}{2} + \cos \frac{\theta}{2}) = \beta$ has a solution, where $\beta$ is a natural number. Then $\beta$ is ______.

Hint:

Always check trigonometric expressions for maximum or minimum when $\beta$ is restricted to integers.

Answer 1

Correct Answer: 1

Step 1: Expand and simplify the expression. 
\[ \sin \frac{\theta}{2} \left(\sin \frac{\theta}{2} + \cos \frac{\theta}{2}\right) = \sin^2 \frac{\theta}{2} + \sin \frac{\theta}{2} \cos \frac{\theta}{2} \] Step 2: Use trigonometric identities. 
\[ \sin^2 \frac{\theta}{2} = \frac{1 - \cos \theta}{2}, \quad \sin \frac{\theta}{2} \cos \frac{\theta}{2} = \frac{1}{2} \sin \theta \] \[ \Rightarrow \beta = \frac{1 - \cos \theta}{2} + \frac{1}{2} \sin \theta \] Step 3: Simplify further. 
\[ \beta = \frac{1}{2}(1 - \cos \theta + \sin \theta) \] Step 4: Maximum possible value. 
Let $f(\theta) = 1 - \cos \theta + \sin \theta$ To maximize $\beta$, set $\sin \theta - \cos \theta = \sqrt{2}\sin(\theta - 45^\circ)$ \[ \text{Maximum of } f(\theta) = 1 + \sqrt{2} \] \[ \Rightarrow \beta_{max} = \frac{1 + \sqrt{2}}{2} \approx 1.207 \] The only natural number satisfying the condition is 1. 
Step 5: Conclusion. 
\[ \boxed{\beta = 1} \]