Question

The triangle PQR of which the angles P, Q, R satisfy \(\cos P = \sin Q = 2 \sin R\) is

• A. equilateral
• B. right angled
• C. any triangle
• D. isosceles

Hint:

Use sine rule: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\).

Answer 1

The Correct Option is D

To determine the type of triangle PQR, we are given the following relationships for the angles:

• \(\cos P = \sin Q = 2 \sin R\)

Our task is to use these trigonometric identities to find the nature of triangle PQR. Let's analyze each step:

1. Since \(\cos P = \sin Q\), by the co-function identity in trigonometry, we know that \(\cos P = \sin(90^\circ - P)\). Thus, we can say \(Q = 90^\circ - P\) or \(P + Q = 90^\circ\).
2. Let us substitute the first equation into the second: Since \(\cos P = 2 \sin R\), we have:
   • \(\sin Q = 2 \sin R\)
   • As per the angle sum property, \(P + Q + R = 180^\circ\).
3. Given the expression \(\sin Q = 2 \sin R\), and knowing that \(\sin(90^\circ - P) = 2 \sin R\), the angles should adjust such that these relationships hold true in the context of a single triangle.
4. One possible scenario matching these conditions is that triangle PQR might be isosceles, where two angles are equal, leading to specific trigonometric identities like sine and cosine to maintain their proportionalities. This aligns with the condition \(\cos P = \sin Q = 2 \sin R\).

After analyzing the given conditions and derived equations, it suggests that triangle PQR is an isosceles triangle to satisfy the equations provided.

Therefore, the correct answer is that triangle PQR is isosceles.