Question

pls help me in this question


sin10° + sin 20° + sin30°+....+sin360° =


Options are


a) 1 b) 0 c) -1 d) infinity​

Answer 1

B] 0

answer is zero.

= sin 10 + sin20 + sin30 + sin40 +...sin180 + sin(360-170) + ……...+ sin(360-40) + sin(360-30) + sin(360-20) + sin360-10) + sin360

sin360-x = - sin x

here all the terms cancled out

so, sin10 + sin20 + sin30 +....+sin 360 = 0

Answer 2

★ Concept :-

Here the concept of Trigonometric Formulas has been used. We see that we are given an expression and we need to find the solution that is we need to find the value of this. So firstly we can expand this expression to different terms. Then by applying Trigonometric Values, we can find the answer.

Let's do it !!

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★ Formula Used :-

$\;\boxed{\sf{\pink{\sin(360^{\circ}\:-\:\theta)\;=\;\bf{-\:\sin\theta}}}}$

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★ Solution :-

Given,

• sin 10° + sin 20° + sin 30° + ... + sin 360° = ?

This can be written as ,

→ sin 10° + sin 20° + sin 30° + ... + sin 170° + sin 180° + sin 190° + ... + sin 340° + sin 350° + sin 360°

Now on applying Trigonometric Identity,

➜ sin(360° - 350°) + sin(360° - 340°) + sin(360° - 330°) + ... + sin(360° - 190°) + sin 180° + sin 190° + ... + sin 340° + sin 350° + sin 360°

• Since, sin(360° - 350°) = - sin 10°

• sin(360° - 340°) = - sin 20°

• sin(360° - 330°) = - sin 30°

• sin(360° - 190°) = - sin 170°

We know that,

$\;\sf{\rightarrow\;\;\sin(360^{\circ}\:-\:\theta)\;=\;\bf{-\:\sin\theta}}$

On applying this formula here, we get

➜ - sin 350° - sin 340° - sin 330° + ... + (- sin 190°) + sin 180° + sin 190° + ... + sin 340° + sin 350° + sin 360°

➜ - sin 350° - sin 340° - sin 330° + ... - sin 190° + sin 180° + sin 190° + ... + sin 340° + sin 350° + sin 360°

Cancelling out the -ve and +ve terms, we get

➜ sin 180° + sin 360°

(since ... terms will cancel out each other)

We know that,

• sin 180° = 0

• sin 360° = 0

➜ 0 + 0

➜ 0

This is the required answer.

So the correct option is option b) 0.

$\;\underline{\boxed{\tt{Required\;\;Answer\;=\;\bf{\purple{0}}}}}$

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★ More to know :-

• cos² x + sin² x = 1

• 1 + tan² x = sec² x

• 1 + cot² x = cosec² x

• cos (-x) = cos x

• sin (-x) = - sin x

• cos (π - x) = - cos x

• sin (π - x) = sin x

• cos (π + x) = - cos x

• sin (π + x) = - sin x

• cos (2π - x) = cos x