What is (p^{2}+q^{2}+r^{2}) equal to?
Consider the following for the three (03) items that follow : Let p=\sin 35^{\circ}, q=\sin 25^{\circ} and r=\sin(-95^{\circ}).
- A. 1/2
- B. 1
- C. 3/2 ✓
- D. 2
Correct Answer: C. 3/2
Explanation
Evaluated in the previous question, p^2+q^2+r^2 = \sin^2 35^{\circ} + \sin^2 25^{\circ} + \sin^2 95^{\circ}. Using \sin^2 A + \sin^2 B = 1 - \cos(A+B)\cos(A-B), \sin^2 35^{\circ} + \sin^2 25^{\circ} = 1 - \cos 60^{\circ} \cos 10^{\circ} = 1 - \frac{1}{2}\cos 10^{\circ}. Since \sin^2 95^{\circ} = \cos^2 5^{\circ} = \frac{1+\cos 10^{\circ}}{2}, the sum is 1 - \frac{1}{2}\cos 10^{\circ} + \frac{1}{2} + \frac{1}{2}\cos 10^{\circ} = 3/2.