Question

$2 \sin \left(\theta+\frac{\pi}{3}\right)=\cos \left(\theta-\frac{\pi}{6}\right)$, then $\tan \theta=$

• A. $-\frac{1}{\sqrt{3}}$
• B. $-\sqrt{3}$
• C. $\sqrt{3}$
• D. $\frac{1}{\sqrt{3}}$

Hint:

Alternatively, use the co-function identity: $\cos\left(\theta - \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{2} - (\theta - \frac{\pi}{6})\right) = \sin\left(\frac{2\pi}{3} - \theta\right)$. You can then expand this to avoid dealing with both sines and cosines initially!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a trigonometric equation involving sum and difference angles and are asked to solve for the value of $\tan \theta$.

Step 2: Key Formula or Approach:
We need to expand the terms using the standard trigonometric addition formulas:
$$\sin(A + B) = \sin A \cos B + \cos A \sin B$$ $$\cos(A - B) = \cos A \cos B + \sin A \sin B$$

Step 3: Detailed Explanation:
Apply the addition formulas to both sides of the equation:
$$2 \left( \sin \theta \cos\frac{\pi}{3} + \cos \theta \sin\frac{\pi}{3} \right) = \cos \theta \cos\frac{\pi}{6} + \sin \theta \sin\frac{\pi}{6}$$ Substitute the known exact values for the trigonometric ratios ($\cos\frac{\pi}{3} = \frac{1}{2}$, $\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$, $\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}$, $\sin\frac{\pi}{6} = \frac{1}{2}$):
$$2 \left( \sin \theta \left(\frac{1}{2}\right) + \cos \theta \left(\frac{\sqrt{3}}{2}\right) \right) = \cos \theta \left(\frac{\sqrt{3}}{2}\right) + \sin \theta \left(\frac{1}{2}\right)$$ Distribute the $2$ on the left side:
$$\sin \theta + \sqrt{3}\cos \theta = \frac{\sqrt{3}}{2}\cos \theta + \frac{1}{2}\sin \theta$$ Group the sine terms on the left and the cosine terms on the right:
$$\sin \theta - \frac{1}{2}\sin \theta = \frac{\sqrt{3}}{2}\cos \theta - \sqrt{3}\cos \theta$$ $$\frac{1}{2}\sin \theta = -\frac{\sqrt{3}}{2}\cos \theta$$ Multiply both sides by $2$:
$$\sin \theta = -\sqrt{3}\cos \theta$$ Divide both sides by $\cos \theta$:
$$\frac{\sin \theta}{\cos \theta} = -\sqrt{3}$$ $$\tan \theta = -\sqrt{3}$$

Step 4: Final Answer:
The value of $\tan \theta$ is $-\sqrt{3}$, which corresponds to option (B).