\(\sin^2 5^\circ+\sin^2 10^\circ+\sin^2 15^\circ+\ldots+\sin^2 90^\circ\) value of:
\(9\dfrac{1}{2}\)
The series is \(\sin^2(5k^\circ)\) for \(k=1,2,\ldots,18\) (angles \(5^\circ,10^\circ,\ldots,90^\circ\)), i.e. 18 terms.
Use the pairing identity \(\sin^2\theta+\sin^2(90^\circ-\theta)=\sin^2\theta+\cos^2\theta=1\). Pair the terms: \((5^\circ,85^\circ),(10^\circ,80^\circ),(15^\circ,75^\circ),(20^\circ,70^\circ),(25^\circ,65^\circ),(30^\circ,60^\circ),(35^\circ,55^\circ),(40^\circ,50^\circ)\) — that is 8 pairs, each summing to 1.
The remaining unpaired terms are \(45^\circ\) and \(90^\circ\): \(\sin^2 45^\circ=\dfrac12\) and \(\sin^2 90^\circ=1\).
Total sum \(=8(1)+\dfrac12+1=9\dfrac12\).
Similar Questions
The value of \(\cos^3(\pi/8)\cdot\cos(3\pi/8) + \sin^3(\pi/8)\cdot\sin(3\pi/8)\) is:
Value of \(\cot\left(\dfrac{\pi}{20}\right)\cot\left(\dfrac{3\pi}{20}\right)\cot\left(\dfrac{5\pi}{20}\right)\cot\left(\dfrac{7\pi}{20}\right)\cot\left(\dfrac{9\pi}{20}\right)\)
The average of \(2\sin 2^\circ,4\sin 4^\circ,6\sin 6^\circ,\ldots,180\sin 180^\circ\) is:
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