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NDA 2014
Exam
NDA
Year
2014
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2014 · Mathematics · Trigonometric Ratios & Identities

What is 20^ ^ + 100^ ^ + 140^ ^ equal to?

What is cos⁡20∘+cos⁡100∘+cos⁡140∘\cos 20^{{}^{\circ}}+\cos 100^{{}^{\circ}}+\cos 140^{{}^{\circ}} equal to?

Options

  1. A2
  2. B1
  3. C1 / 2
  4. D0

Correct answer

D. 0

Step-by-step solution

cos⁡20∘+cos⁡100∘+cos⁡140∘=2cos⁡120∘2⋅cos⁡80∘2+cos⁡140∘=2cos⁡60∘⋅cos⁡40∘+cos⁡140∘=2×12×cos⁡40∘+cos⁡140∘=cos⁡40∘+cos⁡140∘=cos⁡40∘+cos⁡(180∘−40∘)=cos⁡40∘−cos⁡40∘=0.\begin{aligned} & \cos 20^{{}^{\circ}}+\cos 100^{{}^{\circ}}+\cos 140^{{}^{\circ}} \\ & =2 \cos \frac{120^{{}^{\circ}}}{2} \cdot \cos \frac{80^{{}^{\circ}}}{2}+\cos 140^{{}^{\circ}} \\ & =2 \cos 60^{{}^{\circ}} \cdot \cos 40^{{}^{\circ}}+\cos 140^{{}^{\circ}} \\ & =2 \times \frac{1}{2} \times \cos 40^{{}^{\circ}}+\cos 140^{{}^{\circ}} \\ & =\cos 40^{{}^{\circ}}+\cos 140^{{}^{\circ}} \\ & =\cos 40^{{}^{\circ}}+\cos \left(180^{{}^{\circ}}-40^{{}^{\circ}}\right) \\ & =\cos 40^{{}^{\circ}}-\cos 40^{{}^{\circ}}=0.\end{aligned}
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