Question

The value of sin²5° + sin²10° + sin²15° + ... + sin²85° + sin²90° is

Options

• 7

• 8

• 9.5

• 10

Answer 1

9.5

We have: 

\[ \sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + . . . + \sin^2 85^\circ + \sin^2 90^\circ\]

\[ = \sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + . . . + \sin^2 \left( 90^\circ - 10^\circ \right) + \sin^2 \left( 90^\circ - 5^\circ \right) + \sin^2 90^\circ\]

\[ = \sin^2 5^\circ + \sin^2 10^\circ + \sin^2 15^\circ + . . . + \cos^2 10^\circ + \cos^2 5^\circ + \sin^2 90^\circ\]

\[ = \left( \sin^2 5^\circ + \cos^2 5^\circ \right) + \left( \sin^2 10^\circ + \cos^2 10^\circ \right) + + \left( \sin^2 15^\circ + \cos^2 15^\circ \right)\]

\[ + \left( \sin^2 20^\circ + \cos^2 20^\circ \right) + \left( \sin^2 25^\circ + \cos^2 25^\circ \right) + \left( \sin^2 30^\circ + \cos^2 30^\circ \right) \]

\[ + \left( \sin^2 35^\circ + \cos^2 35^\circ \right) + \left( \sin^2 40^\circ + \cos^2 40^\circ \right) + \sin^2 45^\circ + \sin^2 90^\circ\]

\[ = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + \left( \frac{1}{\sqrt{2}} \right)^2 + \left( 1 \right)^2 \left[ \because \sin^2 \theta + \cos^2 \theta = 1 \right]\]

\[ = 8 + \frac{1}{2} + 1\]

\[ = 9 . 5\]