Question

The value of sin2 +sin4 +sin 6 + ...............+sin88 +sin90 is ............

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Answer 1

$\sin 2$ + $\sin 4$ + $\sin 6$ + ...............+ $\sin 88$ + $\sin 90$ = 23

Step-by-step explanation:

We have,

$\sin^2 2$ + $\sin^2 4$ + $\sin^2 6$ + ...............+ $\sin^2 88$ + $\sin^2 90$

To find, $\sin^2 2$ + $\sin^2 4$ + $\sin^2 6$ + ...............+ $\sin^2 88$ + $\sin^2 90$= ?

Total number of terms = $\dfrac{90-2}{2} +1$

= 44 + 1 = 45

∴ $\sin 2$ + $\sin 4$ + $\sin 6$ + ...............+ $\sin 88$ + $\sin 90$

= ($\sin^2 2$ + $\sin^2 88$) + ($\sin^2 4$ + $\sin^2 86)$ +  ($\sin^2 6$ + $\sin^2 84)$ + ..............up to 22 times + $\sin^2 90$

= ($\sin^2 2$ + $\sin^2 (90-2)$) + ($\sin^2 4$ + $\sin^2 (90-4)$) +  ($\sin^2 6$ + $\sin^2 (90-6))$ + ..............up to 22 times + $\sin^2 90$

Using the trigonometric identity,

$\cos A$ = $\sin (90-A)$

= ($\sin^2 2$ + $\cos^2 2$) + ($\sin^2 4$ + $\cos^2 4$) +  ($\sin^2 6$ + $\cos^2 6$) + ..............up to 22 times + 1

Using the trigonometric identity,

$\sin^2 A$ + $\cos^2 A$ = 1

= (1 + 1 + 1 + ....... 22 times) + 1

= 22 + 1

= 23

∴ $\sin 2$ + $\sin 4$ + $\sin 6$ + ...............+ $\sin 88$ + $\sin 90$ = 23

Answer 2

Answer:

Sin8 sin10 etc In the two times tables.

Step-by-step explanation:

You can count up in the times table of 2.