Trigonometry & Inverse Trigonometry
Tangent Addition Formula
Grade 11
Question:
<p>We have <span style='display: inline-block;'>tan(15°) + tan(30°) = –p</span> and <span style='display: inline-block;'>tan(15°) + tan(30°) = –p</span>. Given that tan(45°) = 1, find the value of <span style='display: inline-block;'>(2 + q – p)</span> where <span style='display: inline-block;'>q – p = 1</span>.</p>
Step-by-Step Solution
Key Concept: Use the tangent addition formula and the relationship between angles to establish equations involving p and q.
<p><strong>Step 1:</strong> We have tan(45°) = 1</p><p><strong>Step 2:</strong> Using the tangent addition formula: <span style='display: inline-block;'>tan(30° + 15°) = \frac{\tan(30°) + \tan(15°)}{1 - \tan(30°)\tan(15°)} = 1</span></p><p><strong>Step 3:</strong> Therefore: <span style='display: inline-block;'>\frac{-p}{1 - q} = 1</span></p><p><strong>Step 4:</strong> This gives us: <span style='display: inline-block;'>-p = 1 - q</span></p><p><strong>Step 5:</strong> Rearranging: <span style='display: inline-block;'>q - p = 1</span></p><p><strong>Step 6:</strong> Therefore: <span style='display: inline-block;'>2 + q - p = 2 + 1 = 3</span></p><p>∴ The value of (2 + q – p) is 3.</p>
Correct Answer: 3