Trigonometry
Sum of weighted sines series
MJAT_TS2_P1
Grade 12
Question:
Let $E = 2\sin 2° + 4\sin 4° + 6\sin 6° + \cdots + 178\sin 178°$. Then the value of $\dfrac{E}{\cot 1°}$ is equal to:
Step-by-Step Solution
Key Concept: Let $S = E$. Write $S$ also by reversing using $\sin\theta=\sin(180°-\theta)$: $S = 178\sin 2° + 176\sin 4° + \cdots$. Adding: $2S = 180(\sin 2°+\sin 4°+\cdots+\sin 178°)$. So $S=90\sum_{k=1}^{89}\sin(2k°)$.
$2S = 180\sum_{k=1}^{89}\sin(2k°) = 180\cdot\cot 1°$. So $S = 90\cot 1°$. The value asked is $90$.
Correct Answer: 90