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

Concept:
Use the identity: \[ \sin^2\theta + \sin^2(90^\circ-\theta)=\sin^2\theta+\cos^2\theta=1 \] So we pair terms symmetrically. ip

Step 1: Pair the terms.
We have: \[ \sin^2 5^\circ + \sin^2 85^\circ = 1 \] \[ \sin^2 10^\circ + \sin^2 80^\circ = 1 \] and so on. The pairs are: \[ (5^\circ,85^\circ),\ (10^\circ,80^\circ),\ \dots,\ (40^\circ,50^\circ) \] There are \(8\) such pairs, giving: \[ 8 \] ip

Step 2: Handle the remaining terms.
The terms left are: \[ \sin^2 45^\circ = \frac12 \] and \[ \sin^2 90^\circ = 1 \] ip

Step 3: Add all values.
Total sum: \[ 8+\frac12+1=\frac{19}{2} \] ip Hence, the correct answer is:
\[ \boxed{(A)\ \frac{19}{2}} \]