Question ID: #1155
Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y$ and $z$-axes, respectively, is half of the angle $\alpha$ that this line makes with the positive $x$-axis. Then the sum of all possible values of the angle $\beta$ is
- (1) $\frac{3\pi}{4}$
- (2) $\pi$
- (3) $\frac{\pi}{2}$
- (4) $\frac{3\pi}{2}$
Solution:
$$\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$$
$$\beta = \frac{\alpha}{2}, \gamma = \frac{\alpha}{2}$$
$$\cos^2\alpha + \cos^2\left(\frac{\alpha}{2}\right) + \cos^2\left(\frac{\alpha}{2}\right) = 1$$
$$\cos^2\alpha + 2\cos^2\left(\frac{\alpha}{2}\right) = 1$$
$$2\cos^2\left(\frac{\theta}{2}\right) = 1 + \cos\theta$$
$$\cos^2\alpha + 1 + \cos\alpha = 1$$
$$\cos^2\alpha + \cos\alpha = 0$$
$$\cos\alpha(\cos\alpha + 1) = 0$$
$$\cos\alpha = 0 \Rightarrow \alpha = \frac{\pi}{2}$$
$$\cos\alpha = -1 \Rightarrow \alpha = \pi$$
$$\beta = \frac{\alpha}{2}$$
$$\text{For } \alpha = \frac{\pi}{2} \Rightarrow \beta = \frac{\pi}{4}$$
$$\text{For } \alpha = \pi \Rightarrow \beta = \frac{\pi}{2}$$
$$\text{Sum} = \frac{\pi}{4} + \frac{\pi}{2}$$
$$\text{Sum} = \frac{3\pi}{4}$$
Ans. (1)
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