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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2016 · Mathematics · Trigonometric Ratios & Identities

What is ((θ)/(2))equal to?

What is sin⁡(θ2)equalto?\sin \left(\frac{\theta}{2}\right)equal to?

Options

  1. A∣a^−b^∣2\frac{|\hat{a}-\hat{b}|}{2}
  2. B∣a^+b^∣2\frac{|\hat{\mathrm{a}}+\hat{\mathrm{b}}|}{2}
  3. C∣a^−b^∣4\frac{|\hat{a}-\hat{b}|}{4}
  4. D∣a^+b^∣4\frac{|\hat{a}+\hat{b}|}{4}

Correct answer

A. ∣a^−b^∣2\frac{|\hat{a}-\hat{b}|}{2}

Step-by-step solution

∣a^−b^∣2=(a^−b^)⋅(a^−b^)=a^⋅a^−b^⋅a^−a^⋅b^+b^⋅b^=∣a^∣2−2∣a^∣∣b^∣cos⁡θ+∣b^∣2=2−2cos⁡θ=2(1−cos⁡θ)∣a^−b^∣2=2⋅2sin⁡2θ2sin⁡θ2=∣a^−b^∣2\begin{aligned}|\hat{a}-\hat{b}|^2 & =(\hat{a}-\hat{b}) \cdot(\hat{a}-\hat{b}) \\ & =\hat{a} \cdot \hat{a}-\hat{b} \cdot \hat{a}-\hat{a} \cdot \hat{b}+\hat{b} \cdot \hat{b} \\ & =|\hat{a}|^2-2|\hat{a}||\hat{b}| \cos \theta+|\hat{b}|^2 \\ & =2-2 \cos \theta \\ & =2(1-\cos \theta) \\ |\hat{a}-\hat{b}|^2 & =2 \cdot 2 \sin ^2 \frac{\theta}{2} \\ \sin \frac{\theta}{2}= & \frac{|\hat{a}-\hat{b}|}{2}\end{aligned}
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