Trigonometry & Inverse Trigonometry
Sine Rule and properties of triangles
Grade 11
Question:
<p>In triangle ABC, the ratio \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\) is always equal to (All symbols used have usual meaning in a triangle.)</p>
<p>(a) 2R, where R is the circumradius</p>
<p>(b) \(\frac{2A}{a^2+b^2+c^2}\), where A is the area of the triangle</p>
<p>(c) \(\frac{2}{3}(a^2+b^2+c^2)^{1/2}\)</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: The Sine Rule directly states that the ratio of a side to the sine of its opposite angle is constant and equals the diameter of the circumscribed circle.
<p><strong>Solution:</strong></p><p>This is the Sine Rule (Law of Sines) in trigonometry.</p><p>The ratio $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$</p><p>where R is the circumradius of triangle ABC.</p><p>This fundamental law states that the ratio of any side to the sine of its opposite angle equals twice the circumradius.</p><p>∴ Answer is (a) 2R</p>
Correct Answer: a