Question

If \[ \cos\alpha+\cos\beta+\cos\gamma=0 \] and \[ \sin\alpha+\sin\beta+\sin\gamma=0, \] then which of the following is true?

• A. \(\alpha+\beta+\gamma=\pi\)
• B. \(\alpha+\beta+\gamma=2\pi\)
• C. \(\alpha,\beta,\gamma\) are angles of a triangle
• D. One of the angles differs from another by \(180^\circ\)

Hint:

Whenever sine and cosine sums appear simultaneously, combine them using Euler's formula. Many trigonometric identities become elegant vector problems in the complex plane.

Answer 1

The Correct Option is B

Concept: Using Euler's formula, \[ e^{i\theta} = \cos\theta+i\sin\theta, \] the given equations can be combined into a single complex equation.

Step 1: Combine the two relations Given \[ \cos\alpha+\cos\beta+\cos\gamma=0 \] and \[ \sin\alpha+\sin\beta+\sin\gamma=0. \] Adding the imaginary parts, \[ e^{i\alpha}+e^{i\beta}+e^{i\gamma}=0. \]

Step 2: Geometrical interpretation The numbers \[ e^{i\alpha}, \quad e^{i\beta}, \quad e^{i\gamma} \] lie on the unit circle. Their vector sum is zero. Three unit vectors can add to zero only when they are equally spaced by \(120^\circ\).

Step 3: Determine the sum Hence the arguments are of the form \[ \theta, \quad \theta+\frac{2\pi}{3}, \quad \theta+\frac{4\pi}{3}. \] Their sum is \[ 3\theta+2\pi. \] Modulo \(2\pi\), the standard relation obtained is \[ \boxed{\alpha+\beta+\gamma=2\pi}. \]