Trigonometry & Inverse Trigonometry
Triangle formulas
Grade 11
Question:
<p>In any triangle ABC, the value of $\frac{r_1 + r_2}{1 + \cos C}$ is equal to (where notation have their usual meaning):</p>
<p>(a) $2R$</p>
<p>(b) $2r$</p>
<p>(c) $R$</p>
<p>(d) $\frac{2R^2}{r}$</p>
Step-by-Step Solution
Key Concept: Use exradii formulas and the half-angle formula for cosine combined with standard triangle relations.
<p>Using the formulas: $r_1 = \frac{\Delta}{s-a}$, $r_2 = \frac{\Delta}{s-b}$, and $1 + \cos C = 2\cos^2(\frac{C}{2})$.</p><p>After algebraic manipulation using standard triangle identities, $\frac{r_1 + r_2}{1 + \cos C} = 2R$.</p>
Correct Answer: A