Question

Consider the unit circle with centre $O$ given below

Here, $Q A P$ is tangent to the circle at $A$. Also, the point $A$ is at real number zero. $A P$ and $A Q$ are representing positive real numbers and negative real numbers respectively on this line. On the basis of above information, choose the statement which is incorrect?

• A. $A P$ is in the anti-clockwise direction along the circle
• B. $A Q$ is in the clockwise direction along the circle
• C. Every real number on line $P Q$ will correspond to a radian mcasure and converscly
• D. The length an arc of the circle will give the degree measure of the angle subtended by the arc at the centre

Answer 1

Answer: (D)

This problem is based on relation between radian and real numbers which is given below.

Consider, the unit circle with centre $O$. Let $A$ be any point on the circle. Consider, $O A$ as initial side of an angle. Then, the length of an arc of the circle will give the radian measure of the angle which the arc will subtend at the centre of the circle. Consider, the line $P A Q$ which is tangent to the circle at $A$.
Let the point $A$ represent the real number zero, AP represents positive real number and $A Q$ represents negative real numbers. If we rope the line $A P$ in the anti-clockwise direction along the circle and $A Q$ in the clockwise direction, then every real number will correspond to a radian measure and conversely. Thus, radian measures and real numbers can be considered as one and the same. So, we can conclude that the length of an arc of the circle gives the radian measure of the angle subtended by the arc at the centre.