Question

A ray makes angles \(\frac{\pi}{3}\) and \(\frac{\pi}{4}\) with \(Y\)-axis and \(Z\)-axis respectively. Then the value of the sine of the angle made by the ray with \(X\)-axis is

• A. \(\frac{\sqrt{3}}{2}\)
• B. \(\frac{1}{2}\)
• C. \(\frac{1}{\sqrt{2}}\)
• D. \(1\)

Hint:

If a line makes angles \(\alpha,\beta,\gamma\) with the coordinate axes, then always use \[ \cos^2\alpha+\cos^2\beta+\cos^2\gamma=1. \]

Answer 1

The Correct Option is A

Step 1: Use the direction cosine identity.
If a ray makes angles \(\alpha, \beta, \gamma\) with X, Y, Z axes: \(\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1\). Given \(\beta=\pi/3,\; \gamma=\pi/4\).

Step 2: Solve for \(\cos\alpha\).
\[\cos^2\alpha = 1-\cos^2(\pi/3)-\cos^2(\pi/4) = 1-\frac{1}{4}-\frac{1}{2} = \frac{1}{4}\] So \(\cos\alpha = \pm\dfrac{1}{2}\).

Step 3: Find \(\sin\alpha\).
\(\sin^2\alpha = 1-\cos^2\alpha = 1-\dfrac{1}{4} = \dfrac{3}{4}\). So \(\sin\alpha = \dfrac{\sqrt{3}}{2}\).
\[ \boxed{\dfrac{\sqrt{3}}{2}} \]