The value of sin 2 5 ∘ + sin 2 10 ∘ + sin 2 15 ∘ + .............. + sin 2 85 ∘ + sin 2 90 ∘ is equal to Options: A . &nbsp 7 B . &nbsp 8 C . &nbsp 9 D . &nbsp 19 2

Answer: Option D : D Given expression is sin 2 5 ∘ + sin 2 10 ∘ + sin 2 15 ∘ + . . . . . . . . . . . . . . + sin 2 85 ∘ + sin 2 90 ∘ . We know that sin 90 ∘ = 1 or sin 2 90 ∘ = 1 Similarly, sin 45 ∘ = 1 √ 2 or sin 2 45 ∘ = 1 2 and the angles are in A.P. of 18 terms. We also know that sin 2 85 ∘ = [ sin ( 90 ∘ − 5 ∘ ) ] 2 = cos 2 5 ∘ . Therefore from the complementary rule, we find sin 2 5 ∘ + sin 2 85 ∘ = sin 2 5 ∘ + cos 2 5 ∘ = 1 Therefore, sin 2 5 ∘ + sin 2 10 ∘ + sin 2 15 ∘ + . . . . . . . . . . . . . + sin 2 85 ∘ + sin 2 90 ∘ = ( 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 ) + 1 + 1 2 = 9 1 2 .

The value of sin25∘+sin210∘+sin215∘+.............. +sin2

Discussion Forum : Trigonometric Functions

Question -

The value of ${sin}^{2} 5^{\circ} + {sin}^{2} 10^{\circ} + {sin}^{2} 15^{\circ} +$ ..............

$+ {sin}^{2} 85^{\circ} + {sin}^{2} 90^{\circ}$ is equal to

Options:

A .

7

B .

8

C .

9

D .

\frac{19}{2}

Answer: Option D
:
D

Given expression is

${sin}^{2} 5^{\circ} + {sin}^{2} 10^{\circ} + {sin}^{2} 15^{\circ} + . . . . . . . . . . . . . . + {sin}^{2} 85^{\circ} + {sin}^{2} 90^{\circ}$ .

We know that $sin 90^{\circ} = 1 or {sin}^{2} 90^{\circ} = 1$

Similarly, $sin 45^{\circ} =$ $\frac{1}{\sqrt{2}}$ or ${sin}^{2} 45^{\circ} = \frac{1}{2}$ and the angles are in A.P. of 18 terms. We also know that

${sin}^{2} 85^{\circ} = [sin (90^{\circ} - 5^{\circ})]^{2} = {cos}^{2} 5^{\circ}$ .

Therefore from the complementary rule, we find ${sin}^{2} 5^{\circ} + {sin}^{2} 85^{\circ} = {sin}^{2} 5^{\circ} + {cos}^{2} 5^{\circ} = 1$

Therefore,

${sin}^{2} 5^{\circ} + {sin}^{2} 10^{\circ} + {sin}^{2} 15^{\circ} + . . . . . . . . . . . . . + {sin}^{2} 85^{\circ} + {sin}^{2} 90^{\circ}$

$= (1 + 1 + 1 + 1 + 1 + 1 + 1 + 1) + 1 + \frac{1}{2} = 9 \frac{1}{2}$ .

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Discussion Forum : Trigonometric Functions

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