Question

If \(θ1,θ2, θ3\) are the angles made by a line with the positive directions of the \(x,y,z\) axes, then the value of \(\cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3\) is

• A. \(1\)
• B. \(-1\)
• C. \(2\)
• D. \(-2\)
• E. \(0\)

Answer 1

The Correct Option is B

Given that \(\theta_1, \theta_2,\) and \(\theta_3\) are the angles made by a line with the positive directions of the \(x, y,\) and \(z\) axes, respectively, we know from the property of direction cosines that:

\[ \cos^2 \theta_1 + \cos^2 \theta_2 + \cos^2 \theta_3 = 1 \]

We are asked to find the value of \(\cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3\). Using the double-angle identity for cosine, \(\cos 2\theta = 2\cos^2 \theta - 1\), we can rewrite the expression as:

\[ \cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3 = 2\cos^2 \theta_1 - 1 + 2\cos^2 \theta_2 - 1 + 2\cos^2 \theta_3 - 1 \]

Simplify the expression:

\[ = 2(\cos^2 \theta_1 + \cos^2 \theta_2 + \cos^2 \theta_3) - 3 \]

Substitute the direction cosines property:

\[ = 2(1) - 3 = -1 \]

Thus, the value of \(\cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3\) is \(-1\).

Answer 2

We know that if \( \theta_1, \theta_2, \theta_3 \) are the angles made by a line with the positive directions of the x, y, and z axes, then

\( \cos^2\theta_1 + \cos^2\theta_2 + \cos^2\theta_3 = 1 \)

Also, we know the trigonometric identity \( \cos 2\theta = 2\cos^2\theta - 1 \). Thus,

\( \cos 2\theta_1 = 2\cos^2\theta_1 - 1 \)

\( \cos 2\theta_2 = 2\cos^2\theta_2 - 1 \)

\( \cos 2\theta_3 = 2\cos^2\theta_3 - 1 \)

Adding the three equations, we get

\( \cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3 = 2(\cos^2\theta_1 + \cos^2\theta_2 + \cos^2\theta_3) - 3 \)

Since \( \cos^2\theta_1 + \cos^2\theta_2 + \cos^2\theta_3 = 1 \), we have

\( \cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3 = 2(1) - 3 \)

\( \cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3 = 2 - 3 = -1 \)

Final Answer: The final answer is \(-1\)