Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>The most general values of \(\theta\) satisfying \(\tan\left(\theta + \frac{3\pi}{4}\right) + \tan\theta = 2\) is/are</p>
<p>(a) \(n\pi \pm \frac{\pi}{3}, n \in \mathbb{I}\)</p>
<p>(b) \(2n\pi + \frac{\pi}{3}, n \in \mathbb{I}\)</p>
<p>(c) \(2n\pi \pm \frac{\pi}{3}, n \in \mathbb{I}\)</p>
<p>(d) \(2n\pi + (-1)^n \frac{\pi}{3}, n \in \mathbb{I}\)</p>
Step-by-Step Solution
Key Concept: Apply the tangent addition formula to expand $\tan(\theta + \frac{3\pi}{4})$ and use the condition that $\tan\frac{3\pi}{4} = -1$.
<p>Using the tangent addition formula and simplification, the general solution is $\theta = n\pi \pm \frac{\pi}{3}$ where $n \in \mathbb{I}$.</p>
Correct Answer: a