Trigonometry & Inverse Trigonometry
Triangle Classification
Grade 11
Question:
<p>If in a △<i>ABC</i>, \(\frac{a^2 - b^2}{a^2 + b^2} = \frac{\sin(A-B)}{\sin(A+B)}\), then the triangle is</p>
<p>(a) Right angled or isosceles</p>
<p>(b) Right angled and isosceles</p>
<p>(c) Equilateral</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use sine rule and trigonometric identities to relate side ratios to angle conditions.
<p>Using sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = 2R$</p><p>LHS: $\frac{a^2 - b^2}{a^2 + b^2}$</p><p>RHS: $\frac{\sin(A-B)}{\sin(A+B)}$</p><p>By sine rule manipulation and product-to-sum formulas, this equation simplifies when either $A = \frac{\pi}{2}$ (right angled) or $a = b$ (isosceles).</p>
Correct Answer: A