Question

A ray makes angles \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \) with Y and Z-axes 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/ray makes angles \(\alpha, \beta, \gamma\) with the x, y, z axes respectively, then \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\).
\(\sin^2\theta + \cos^2\theta = 1\).
The angle made by a ray with an axis is typically \(0 \le \theta \le \pi\), so \(\sin\theta \ge 0\).

Answer 1

The Correct Option is A

Let \(\alpha, \beta, \gamma\) be the angles made by the ray with the positive X, Y, and Z-axes respectively. The direction cosines of the ray are \(l = \cos\alpha\), \(m = \cos\beta\), \(n = \cos\gamma\). We know the identity \(l^2+m^2+n^2 = 1\), which means \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\). Given: Angle with Y-axis, \(\beta = \pi/3\). So, \(\cos\beta = \cos(\pi/3) = 1/2\). Angle with Z-axis, \(\gamma = \pi/4\). So, \(\cos\gamma = \cos(\pi/4) = 1/\sqrt{2}\). Substitute these into the identity: \(\cos^2\alpha + (\cos\beta)^2 + (\cos\gamma)^2 = 1\) \(\cos^2\alpha + (1/2)^2 + (1/\sqrt{2})^2 = 1\) \(\cos^2\alpha + 1/4 + 1/2 = 1\) \(\cos^2\alpha + 1/4 + 2/4 = 1\) \(\cos^2\alpha + 3/4 = 1\) \(\cos^2\alpha = 1 - 3/4 = 1/4\). So, \(\cos\alpha = \pm\sqrt{1/4} = \pm 1/2\). We need to find the sine of the angle made by the ray with the X-axis, which is \(\sin\alpha\). We know \(\sin^2\alpha + \cos^2\alpha = 1\). So, \(\sin^2\alpha = 1 - \cos^2\alpha\). Since \(\cos^2\alpha = 1/4\): \(\sin^2\alpha = 1 - 1/4 = 3/4\). \(\sin\alpha = \pm\sqrt{3/4} = \pm \frac{\sqrt{3}}{2}\). The angle \(\alpha\) made by a ray with an axis is usually taken to be in the range \([0, \pi]\). In this range, \(\sin\alpha \ge 0\). Therefore, \(\sin\alpha = \frac{\sqrt{3}}{2}\). This matches option (a). \[ \boxed{\frac{\sqrt{3}}{2}} \]