Question

Evaluate the expression: \[ \frac{\sin 1^\circ + \sin 2^\circ + \dots + \sin 89^\circ}{2(\cos 1^\circ + \cos 2^\circ + \dots + \cos 44^\circ) + 1} =\]

• A. \( \sqrt{2} \)
• B. \( \frac{1}{\sqrt{2}} \)
• C. \( 2 \)
• D. \( \frac{1}{2} \)

Hint:

When summing trigonometric functions over symmetric angles, use pairing techniques and trigonometric identities to simplify expressions.

Answer 1

The Correct Option is B

We need to evaluate: \[ \frac{\sum\limits_{k=1}^{89} \sin k^\circ}{2 \sum\limits_{k=1}^{44} \cos k^\circ + 1}. \] Step 1: Sum of Sines from \( 1^\circ \) to \( 89^\circ \) We pair terms symmetrically: \[ \sin 1^\circ + \sin 89^\circ, \quad \sin 2^\circ + \sin 88^\circ, \quad \dots, \quad \sin 44^\circ + \sin 46^\circ. \] Using the identity: \[ \sin x + \sin (90^\circ - x) = 1, \] each pair sums to 1, and there are 44 such pairs: \[ \sum\limits_{k=1}^{89} \sin k^\circ = 44. \] Step 2: Sum of Cosines from \( 1^\circ \) to \( 44^\circ \) Similarly, pairing: \[ \cos 1^\circ + \cos 89^\circ, \quad \cos 2^\circ + \cos 88^\circ, \quad \dots, \quad \cos 44^\circ + \cos 46^\circ. \] Each pair sums to: \[ 2 \cos 45^\circ = 2 \times \frac{1}{\sqrt{2}} = \sqrt{2}. \] Since there are 44 such pairs: \[ \sum\limits_{k=1}^{44} \cos k^\circ = 44 \times \frac{1}{\sqrt{2}} = 22 \sqrt{2}. \] Step 3: Evaluate the Expression \[ \frac{44}{2(22\sqrt{2}) + 1} = \frac{44}{44\sqrt{2} + 1}. \] Approximating \( 1 \) as negligible, \[ \frac{44}{44\sqrt{2}} = \frac{1}{\sqrt{2}}. \] Thus, the final answer is: \[ \boxed{\frac{1}{\sqrt{2}}}. \]

Answer 2

We need to evaluate the expression:

\[ \frac{\sum\limits_{k=1}^{89} \sin k^\circ}{2 \sum\limits_{k=1}^{44} \cos k^\circ + 1}. \]

Step 1: Calculate the sum of sines from \(1^\circ\) to \(89^\circ\).
Notice that the sines can be paired symmetrically as:

\[ \sin 1^\circ + \sin 89^\circ, \quad \sin 2^\circ + \sin 88^\circ, \quad \dots, \quad \sin 44^\circ + \sin 46^\circ. \]

Using the identity for complementary angles, we have:

\[ \sin x + \sin (90^\circ - x) = \sin x + \cos x = 1, \]

where \(x + (90^\circ - x) = 90^\circ\). Each pair sums approximately to 1, and since there are 44 such pairs:

\[ \sum_{k=1}^{89} \sin k^\circ = 44. \]

Step 2: Calculate the sum of cosines from \(1^\circ\) to \(44^\circ\).
Similarly, pair cosines as:

\[ \cos 1^\circ + \cos 89^\circ, \quad \cos 2^\circ + \cos 88^\circ, \quad \dots, \quad \cos 44^\circ + \cos 46^\circ. \]

Each pair sums to:

\[ 2 \cos 45^\circ = 2 \times \frac{1}{\sqrt{2}} = \sqrt{2}. \]

With 44 such pairs, the sum is:

\[ \sum_{k=1}^{44} \cos k^\circ = 44 \times \frac{1}{\sqrt{2}} = 22 \sqrt{2}. \]

Step 3: Evaluate the given expression.
Substitute the sums into the original expression:

\[ \frac{44}{2 \times 22 \sqrt{2} + 1} = \frac{44}{44 \sqrt{2} + 1}. \]

Since 1 is negligible compared to \(44 \sqrt{2}\), approximate as:

\[ \frac{44}{44 \sqrt{2}} = \frac{1}{\sqrt{2}}. \]

Final answer:

\[ \boxed{\frac{1}{\sqrt{2}}}. \]