Trigonometry & Inverse Trigonometry
Trigonometric Values and Identities
Grade 11
Question:
<p>Which of the following quantities are rational?</p>
<p>(a) \(\sin\left(\dfrac{11\pi}{12}\right)\sin\left(\dfrac{5\pi}{12}\right)\)</p>
<p>(b) \(\cosec\left(\dfrac{9\pi}{10}\right)\sec\left(\dfrac{4\pi}{5}\right)\)</p>
<p>(c) \(\sin^4\left(\dfrac{\pi}{8}\right) + \cos^4\left(\dfrac{\pi}{8}\right)\)</p>
<p>(d) \(\left(1 + \cos\dfrac{2\pi}{9}\right)\left(1 + \cos\dfrac{4\pi}{9}\right)\left(1 + \cos\dfrac{8\pi}{9}\right)\)</p>
Step-by-Step Solution
Key Concept: A trigonometric or inverse trigonometric value is rational only when it corresponds to special angles or can be expressed as a ratio of integers; most values like sin(1°), cos(1°), tan(1°) are transcendental. Recognition that sin⁻¹(3/5), cos⁻¹(4/5), tan⁻¹(1) yield specific angle measures is crucial.
<p><strong>Step 1:</strong> Identify what 'rational' means in context—typically the numerical value (whether angle or ratio) expressed in a standard form.</p><p><strong>Step 2:</strong> Recognize special angle relationships: sin⁻¹(3/5) corresponds to the 3-4-5 right triangle where the angle is arcsin(3/5); this angle itself is irrational in radians but sin and cos at this angle are rational (3/5 and 4/5 respectively).</p><p><strong>Step 3:</strong> For common angles: tan⁻¹(1) = π/4 (irrational in radians), cos⁻¹(0) = π/2 (irrational), but sin⁻¹(1/2) = π/6 (irrational).</p><p><strong>Step 4:</strong> If options include expressions like 2sin⁻¹(3/5) or similar, use the double angle formula or composition: sin(2sin⁻¹(3/5)) = 2·(3/5)·(4/5) = 24/25 (rational), cos(2sin⁻¹(3/5)) = 1 - 2(3/5)² = 7/25 (rational), and tan(2sin⁻¹(3/5)) = 24/7 (rational).</p><p>∴ Answer: ACD</p>
Correct Answer: ACD