Question

Evaluate the sum: \[ \sin^2 18^\circ + \sin^2 24^\circ + \sin^2 36^\circ + \sin^2 42^\circ + \sin^2 78^\circ + \sin^2 90^\circ + \sin^2 96^\circ + \sin^2 102^\circ + \sin^2 138^\circ + \sin^2 162^\circ. \]

• A. \( \frac{11}{2} \)
• B. \( \frac{9}{2} \)
• C. \( 5 \)
• D. \( 4 \)

Hint:

Use the angle addition formula \(\cos (a + b) = \cos a \cos b - \sin a \sin b\) and the double-angle formula \(\cos 2a = 2 \cos^2 a - 1 = 1 - 2 \sin^2 a\).

Answer 1

The Correct Option is A

A straightforward approach is to evaluate each term numerically (for example, via a calculator or a short computational script) and sum them. Recall that \(\sin^2 \theta = \dfrac{1 - \cos(2\theta)}{2}\). One way to see if the sum simplifies nicely is to convert each term using this identity; however, direct numerical evaluation is simple and sufficient for this finite sum: \[ \sin^2 \theta = \bigl(\sin \theta \bigr)^2. \] Using a calculator or short program: \[ \sin^2(18^\circ) + \sin^2(24^\circ) + \sin^2(36^\circ) + \sin^2(42^\circ) \]\[ + \sin^2(78^\circ) + \sin^2(90^\circ) + \sin^2(96^\circ) + \sin^2(102^\circ) + \sin^2(138^\circ) + \sin^2(162^\circ) \approx 5.5. \] Since \(5.5 = \frac{11}{2}\), the exact value of the sum is \[ \boxed{\frac{11}{2}}. \]