Multiple choice

A variable triangle is inscribed in a circle of radius R. If the rate of change of side is R times the rate of change of the opposite angle, then that angle is

  1. $\pi/6$
  2. $\pi/4$
  3. $\pi/3$
  4. $\pi/2$
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C Correct answer
Explanation

By the Law of Sines, a = 2R sin(A). Differentiating with respect to time, da/dt = 2R cos(A) * dA/dt. Given da/dt = R * dA/dt, we have 2R cos(A) = R, so cos(A) = 1/2. Thus, A = 60 degrees or pi/3.

AI explanation

Using the extended law of sines, the side of a triangle is given by a = 2R sin(A), where R is the circumradius and A is the opposite angle. Differentiating this with respect to the angle gives da/dA = 2R cos(A). The problem states that the rate of change of the side equals R times the rate of change of the opposite angle, meaning da/dA = R. Setting the two equations equal gives 2R cos(A) = R, so cos(A) = 1/2. The angle whose cosine is 1/2 is pi/3.

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