What is sin(θ /2) equal to?

Correct Answer is : ∣a^−b^∣2{ | {a}-{b} | }{2}2∣a^−b^∣

Correct Answer is : ∣a^−b^∣2{ | {a}-{b} | }{2}2∣a^−b^∣

Test Index

UPSC NDA I 2016 Mathematics Solved Paper

Question Numbers: 74-75

Direction: For the next two (2) items that follow:

Let â, b̂ be two unit vectors and θ be the angle between them.

Question : 75 of 120

Marks: +1, -0

What is

sin(\frac{\theta }{2})

equal to?

Solution:

Concept:

Use the dot product identity for unit vectors and the half-angle formula:

1 - \cos\theta = 2\sin^2\frac{\theta}{2}

Explanation:

Given

|\hat{a}| = 1

and

|\hat{b}| = 1

Compute

|\hat{a} - \hat{b}|^2 = |\hat{a}|^2 + |\hat{b}|^2 - 2(\hat{a}\cdot\hat{b}) = 1+1-2\cos\theta = 2(1-\cos\theta)

Substitute

1-\cos\theta = 2\sin^2\frac{\theta}{2}

|\hat{a} - \hat{b}|^2 = 2 \times 2\sin^2\frac{\theta}{2} = 4\sin^2\frac{\theta}{2}

Taking the positive square root (since

\sin\frac{\theta}{2} \ge 0

for

0 \le \theta \le \pi

), we get

\sin\frac{\theta}{2} = \frac{|\hat{a} - \hat{b}|}{2}

Thus, the expression matches option A.

Answer:

\sin\frac{\theta}{2} = \frac{|\hat{a} - \hat{b}|}{2}

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