Trigonometry & Inverse Trigonometry
Trigonometric identities
Grade None
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 angles to complementary/supplementary forms: cos 65° = sin 25°, cos 55° = sin 35°, and cos 175° = -cos 5°. Then use product-to-sum formulas and recognize that cos 55° · cos 65° = cos 55° · sin 25° relates to sin 80°.
<p><strong>Step 1:</strong> Rewrite using complementary angles:</p><p>• p = cos 55° = sin 35°</p><p>• q = cos 65° = sin 25°</p><p>• r = cos 175° = -cos 5°</p><p><strong>Step 2:</strong> Express the given expression:</p><p>$$\frac{1}{p} + \frac{1}{q} + \frac{r}{pq} = \frac{q + p + r}{pq}$$</p><p><strong>Step 3:</strong> Calculate pq using product formula:</p><p>$$pq = \sin 35° \cdot \sin 25° = \frac{1}{2}[\cos(35°-25°) - \cos(35°+25°)]$$</p><p>$$= \frac{1}{2}[\cos 10° - \cos 60°] = \frac{1}{2}[\cos 10° - \frac{1}{2}]$$</p><p><strong>Step 4:</strong> Calculate numerator:</p><p>$$p + q + r = \sin 35° + \sin 25° - \cos 5°$$</p><p>$$= 2\sin 30° \cos 5° - \cos 5° = \cos 5° - \cos 5° = 0$$</p><p>Wait—recalculate: sin 35° + sin 25° = 2 sin 30° cos 5° = 2(½) cos 5° = cos 5°</p><p>So: p + q + r = cos 5° - cos 5° = 0</p><p><strong>Step 5:</strong> Since numerator = 0:</p><p>$$\frac{1}{p} + \frac{1}{q} + \frac{r}{pq} = \frac{0}{pq} = 0$$</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: C