Question

The value of \(\sin^2 5^\circ + \sin^2 10^\circ + \dots + \sin^2 85^\circ + \sin^2 90^\circ =\)

• A. \(19/2\)
• B. \(3/2\)
• C. \(23/2\)
• D. \(21/2\)

Hint:

In sums like \(\sin^2\theta + \sin^2(90^\circ-\theta)\), always pair complementary angles first.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Use \(\sin^2 \theta + \sin^2(90-\theta) = 1\).
Step 2: Key Formula or Approach:
Pair terms: \((5, 85), (10, 80), \dots, (40, 50)\).
Step 3: Detailed Explanation:
There are \(18\) terms. \(8\) pairs make \(8\).
Middle \(\sin^2 45 = 1/2\). Last \(\sin^2 90 = 1\).
Total \(= 8 + 1 + 0.5 = 9.5 = 19/2\).
Step 4: Final Answer:
Sum is \(19/2\).