Question

The general solution of \( \tan \theta + \tan 2\theta = \tan 3\theta \)

• A. \( \theta = (2n + 1) \frac{\pi}{2}, n \in \mathbb{Z} \)
• B. \( \theta = n\pi, n \in \mathbb{Z} \quad \text{or} \quad \theta = p \frac{\pi}{3}, p \in \mathbb{Z} \)
• C. \( \theta = \frac{n\pi}{5}, n \in \mathbb{Z} \)
• D. \( \theta = (2n - 1) \frac{\pi}{3}, n \in \mathbb{Z} \)

Hint:

When solving trigonometric equations, look for patterns such as periodicity and apply trigonometric identities to simplify the equation.

Answer 1

The Correct Option is B

Step 1: Solve the given trigonometric equation.
The given equation is: \[ \tan \theta + \tan 2\theta = \tan 3\theta \] We use the tangent addition formula to simplify: \[ \tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] So, we first express \( \tan 3\theta \) as: \[ \tan 3\theta = \frac{\tan \theta + \tan 2\theta}{1 - \tan \theta \tan 2\theta} \]
Step 2: Analyze the solutions.
By solving this equation, we get: \[ \theta = n\pi, n \in \mathbb{Z} \quad \text{or} \quad \theta = p \frac{\pi}{3}, p \in \mathbb{Z} \]
Step 3: Conclusion.
Thus, the general solution is \( \theta = n\pi, n \in \mathbb{Z} \quad \text{or} \quad \theta = p \frac{\pi}{3}, p \in \mathbb{Z} \), corresponding to option (B).