Question

The value of θ in the range 0 ≤ θ ≤ π/2 which satisfies the equation sin (θ + π/6) = cosθ is equal to

• A. π/6
• B. π/4
• C. π/3
• D. π/8
• E. π/5

Answer 1

The Correct Option is A

Given equation:

\(\sin \left(\theta + \frac{\pi}{6} \right) = \cos \theta\)

Using the identity:

\(\cos \theta = \sin \left(\frac{\pi}{2} - \theta\right)\), we rewrite the equation as:

\(\sin \left(\theta + \frac{\pi}{6} \right) = \sin \left(\frac{\pi}{2} - \theta\right)\)

For \(\sin A = \sin B\), the general solution is:

\(A = B + 2k\pi\) or \(A = \pi - B + 2k\pi\) for some integer \(k\).

Applying this to our equation:

\(\theta + \frac{\pi}{6} = \frac{\pi}{2} - \theta\)

Solving for \(\theta\):

\(\theta + \theta = \frac{\pi}{2} - \frac{\pi}{6}\)

\(2\theta = \frac{\pi}{3}\)

\(\theta = \frac{\pi}{6}\)

Since \(\theta\) lies in the given range \(0 \leq \theta \leq \frac{\pi}{2}\), the valid solution is:

\(\theta = \frac{\pi}{6}\)