Trigonometry & Inverse Trigonometry
Applications of Trigonometry
Grade 11
Question:
<p>A train travelling on one of two intersecting railway lines, subtends at a certain station on the other line, an angle \(\alpha\) when the front of the carriage reaches the junction and an angle \(\beta\) when the end of the carriage reaches it. The two lines are inclined to each other at an angle \(\theta\). Then \(2\cot\theta\) is equal to</p>
<p>(a) \(\tan\alpha - \tan\beta\)</p>
<p>(b) \(\cot\beta - \cot\alpha\)</p>
<p>(c) \(\cot\alpha - \cot\beta\)</p>
<p>(d) \(\cot\alpha + \cot\beta\)</p>
Step-by-Step Solution
Key Concept: Apply trigonometric identities for angles subtended in configurations with intersecting lines to establish the relationship between the angles.
<p><strong>Solution approach:</strong> Two intersecting railway lines $l_1$ and $l_2$ meet at a station. A train carriage on one line subtends angles $\alpha$ and $\beta$ at a station on the other line, measured from the front and rear respectively. The angle between the lines is $\theta$.</p><p>Using trigonometric relationships in the configuration of the two intersecting lines and the angles subtended by the carriage, we can derive that:</p><p>$$2\cot\theta = \cot\alpha - \cot\beta$$</p><p>∴ Answer is (c).</p>
Correct Answer: c