Trigonometry & Inverse Trigonometry
Trigonometric identities and quadrant
Grade 11
Question:
<p>Given \(\tan\theta = \dfrac{-4}{3}\), find \(\sin\theta\).</p>
<p>\(\dfrac{4}{5}\) only</p>
<p>\(\dfrac{4}{5}\) or \(\dfrac{-4}{5}\)</p>
<p>\(\dfrac{-4}{5}\) only</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Use the identity sec²θ = 1 + tan²θ to find cos²θ, then determine the correct sign of sinθ based on the quadrant where tanθ is negative.
<p><strong>Step 1:</strong> Use the identity sec²θ = 1 + tan²θ</p><p>sec²θ = 1 + (-4/3)² = 1 + 16/9 = 25/9</p><p>Therefore, cos²θ = 9/25, so cosθ = ±3/5</p><p><strong>Step 2:</strong> Since tanθ = sinθ/cosθ = -4/3 is negative, sinθ and cosθ have opposite signs.</p><p><strong>Step 3:</strong> If cosθ = 3/5 (positive), then sinθ = tanθ · cosθ = (-4/3)(3/5) = -4/5</p><p>If cosθ = -3/5 (negative), then sinθ = (-4/3)(-3/5) = 4/5</p><p><strong>Step 4:</strong> Both solutions are mathematically valid depending on the quadrant:</p><ul><li><strong>Quadrant II:</strong> sinθ = 4/5, cosθ = -3/5</li><li><strong>Quadrant IV:</strong> sinθ = -4/5, cosθ = 3/5</li></ul><p>If additional constraint specifies a quadrant or range for θ, use that to select the appropriate answer.</p><p>∴ Answer: <strong>B</strong> (Most commonly sinθ = ±4/5, or if B specifies one quadrant: 4/5 for QII or -4/5 for QIV)</p>
Correct Answer: B