Trigonometry & Inverse Trigonometry
Properties of Triangles
Grade 11
Question:
<p>The value of <span class="math">\(\cos A + \cos B + \cos C\)</span> is</p>
<p>(a) <span class="math">\(1 + \frac{r}{R}\)</span></p>
<p>(b) <span class="math">\(2\left(1 + \frac{r}{R}\right)\)</span></p>
<p>(c) <span class="math">\(\frac{1}{2}\left(1 + \frac{r}{R}\right)\)</span></p>
<p>(d) <span class="math">\(\frac{1}{4}\left(1 + \frac{r}{R}\right)\)</span></p>
Step-by-Step Solution
Key Concept: Express each cosine using the cosine rule and then sum them using properties of inradius and circumradius.
<p>Using the projection formula and the relations <span class="math">\(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)</span> and expressing in terms of inradius <span class="math">\(r\)</span> and circumradius <span class="math">\(R\)</span>, we get <span class="math">\(\cos A + \cos B + \cos C = 1 + \frac{r}{R}\)</span>.</p><p>∴ Answer is (a).</p>
Correct Answer: A