QuestionApril 16, 2026

35. 2sin(Theta )/(2)=sinTheta

35. 2sin(Theta )/(2)=sinTheta
35. 2sin(Theta )/(2)=sinTheta

Solution
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Answer

\Theta = 2n\pi,\ n \in \mathbb{Z} Explanation 1. Rewrite \sin\Theta using double-angle formula \sin\Theta = 2\sin\frac{\Theta}{2} \cos\frac{\Theta}{2}. 2. Substitute into equation 2\sin\frac{\Theta}{2} = 2\sin\frac{\Theta}{2} \cos\frac{\Theta}{2}. 3. Simplify Divide both sides by 2\sin\frac{\Theta}{2} (valid when \sin\frac{\Theta}{2} \neq 0), giving 1 = \cos\frac{\Theta}{2}. 4. Solve for \Theta when \sin\frac{\Theta}{2} \neq 0 \cos\frac{\Theta}{2} = 1 \Rightarrow \frac{\Theta}{2} = 2n\pi \Rightarrow \Theta = 4n\pi. 5. Solve for \sin\frac{\Theta}{2} = 0 \frac{\Theta}{2} = n\pi \Rightarrow \Theta = 2n\pi. 6. Combine solutions From steps 4 and 5, solutions are \Theta = 2n\pi.

Explanation

1. Rewrite $\sin\Theta$ using double-angle formula<br /> $\sin\Theta = 2\sin\frac{\Theta}{2} \cos\frac{\Theta}{2}$.<br />2. Substitute into equation<br /> $2\sin\frac{\Theta}{2} = 2\sin\frac{\Theta}{2} \cos\frac{\Theta}{2}$.<br />3. Simplify<br /> Divide both sides by $2\sin\frac{\Theta}{2}$ (valid when $\sin\frac{\Theta}{2} \neq 0$), giving $1 = \cos\frac{\Theta}{2}$.<br />4. Solve for $\Theta$ when $\sin\frac{\Theta}{2} \neq 0$<br /> $\cos\frac{\Theta}{2} = 1 \Rightarrow \frac{\Theta}{2} = 2n\pi \Rightarrow \Theta = 4n\pi$.<br />5. Solve for $\sin\frac{\Theta}{2} = 0$<br /> $\frac{\Theta}{2} = n\pi \Rightarrow \Theta = 2n\pi$.<br /><br />6. Combine solutions<br /> From steps 4 and 5, solutions are $\Theta = 2n\pi$.
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