TS EAMCET · Maths · Trigonometric Ratios & Identities
\(\frac{\sin 1^{\circ}+\sin 2^{\circ}+\ldots+\sin 89^{\circ}}{2\left(\cos 1^{\circ}+\cos 2^{\circ}+\ldots+\cos 44^{\circ}\right)+1}=\)
- A \(2\)
- B \(\frac{1}{\sqrt{2}}\)
- C \(\frac{1}{2}\)
- D \(\sqrt{2}\)
Answer & Solution
Correct Answer
(B) \(\frac{1}{\sqrt{2}}\)
Step-by-step Solution
Detailed explanation
\( \sum_{k=1}^{89} \sin k^{\circ} = (\sin 1^{\circ}+\sin 89^{\circ}) + \ldots + (\sin 44^{\circ}+\sin 46^{\circ}) + \sin 45^{\circ} \) \( \sin k^{\circ}+\sin(90^{\circ}-k^{\circ}) = 2\sin 45^{\circ}\cos(k-45^{\circ}) = \sqrt{2}\cos(45^{\circ}-k^{\circ}) \)…
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