Trigonometry & Inverse Trigonometry
Trigonometric Identities
Grade 11
Question:
<p>If \(p = \cos 55°\), \(q = \cos 65°\) and \(r = \cos 175°\), then the value of \(\dfrac{1}{p} + \dfrac{1}{q} + \dfrac{r}{pq}\) is equal to:</p>
<p>(a) 0</p>
<p>(b) -1</p>
<p>(c) 1</p>
<p>(d) 2</p>
Step-by-Step Solution
Key Concept: Convert cos(175°) = -cos(5°) and use complementary angle relationships (cos 55° = sin 35°, cos 65° = sin 25°) along with sum-to-product formulas to simplify the reciprocal expression.
<p><strong>Step 1:</strong> Recognize angle relationships.</p><p>cos 55° = sin 35°, cos 65° = sin 25°, and cos 175° = -cos 5°</p><p><strong>Step 2:</strong> Combine fractions with common denominator pq.</p><p>$$\frac{1}{p} + \frac{1}{q} + \frac{r}{pq} = \frac{q + p + r}{pq}$$</p><p><strong>Step 3:</strong> Substitute values.</p><p>$$= \frac{\cos 55° + \cos 65° - \cos 5°}{\cos 55° \cdot \cos 65°}$$</p><p><strong>Step 4:</strong> Use sum-to-product on cos 55° + cos 65°.</p><p>$$\cos 55° + \cos 65° = 2\cos 60° \cos 5° = 2 \cdot \frac{1}{2} \cdot \cos 5° = \cos 5°$$</p><p><strong>Step 5:</strong> Simplify numerator.</p><p>$$\cos 5° - \cos 5° = 0$$</p><p><strong>Step 6:</strong> Evaluate.</p><p>$$\frac{0}{\cos 55° \cdot \cos 65°} = 0$$</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: A