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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2022 · Mathematics · Trigonometric Ratios & Identities

What is the value of 0^ ^ + 10^ ^ + 20^ ^ + 30^ ^ + .+ 360^ ^ ?

What is the value of sin⁡0∘+sin⁡10∘+sin⁡20∘+sin⁡30∘+….+sin⁡360∘\sin 0^{{}^{\circ}}+\sin 10^{{}^{\circ}}+\sin 20^{{}^{\circ}}+\sin 30^{{}^{\circ}}+\ldots.+\sin 360^{{}^{\circ}} ?

Options

  1. A-1
  2. B0
  3. C1
  4. D2

Correct answer

B. 0

Step-by-step solution

sin⁡0∘+sin⁡10∘+sin⁡20∘+sin⁡30∘+….+sin⁡360∘=(sin⁡0∘+sin⁡360∘)+(sin⁡10∘+sin⁡350∘)+(sin⁡20∘+sin⁡340∘)+…+sin⁡180∘=2sin⁡180∘⋅cos⁡180∘+2sin⁡180∘⋅cos⁡170∘+2sin⁡180∘⋅cos⁡160∘)+…+sin⁡180∘=0+0+0….+0[∵sin⁡180∘=0]=0\begin{aligned} & \sin 0^{{}^{\circ}}+\sin 10^{{}^{\circ}}+\sin 20^{{}^{\circ}}+\sin 30^{{}^{\circ}}+\ldots.+\sin 360^{{}^{\circ}} \\ & =\left(\sin 0^{{}^{\circ}}+\sin 360^{{}^{\circ}}\right)+\left(\sin 10^{{}^{\circ}}+\sin 350^{{}^{\circ}}\right)+\left(\sin 20^{{}^{\circ}}+\right. \\ & \left.\sin 340^{{}^{\circ}}\right)+\ldots+\sin 180^{{}^{\circ}} \\ & =2 \sin 180^{{}^{\circ}} \cdot \cos 180^{{}^{\circ}}+2 \sin 180^{{}^{\circ}} \cdot \cos 170^{{}^{\circ}}+ \\ & \left.2 \sin 180^{{}^{\circ}} \cdot \cos 160^{{}^{\circ}}\right)+\ldots+\sin 180^{{}^{\circ}} \\ & =0+0+0 \ldots.+0 \quad\left[\because \sin 180^{{}^{\circ}}=0\right] \\ & =0\end{aligned}
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