Question

q1. area by two radii with general angles
points a and b lie on the unit circle at angles 40° and 140°. what is the area of triangle oab?
a. \\(\frac{1}{2} \sin 100°\\)
b. \\(\frac{1}{2} \sin 40°\\)
c. \\(\sin 50°\\)
d. \\(\sin 20°\\)
e. \\(\frac{1}{2} \sin 80°\\)

Explanation:

Step1: Recall the formula for the area of a triangle with two sides and included angle

The formula for the area of a triangle with two sides \(a\) and \(b\) and included angle \(\theta\) is \(A=\frac{1}{2}ab\sin\theta\). In the unit circle, the radius \(OA = OB=1\) (since \(A\) and \(B\) lie on the unit circle).

Step2: Find the included angle between \(OA\) and \(OB\)

The angles of points \(A\) and \(B\) with respect to the positive \(x -\)axis are \(40^{\circ}\) and \(140^{\circ}\) respectively. The included angle \(\angle AOB=140^{\circ}- 40^{\circ}=100^{\circ}\).

Step3: Substitute into the area formula

Using the formula \(A = \frac{1}{2}ab\sin\theta\) with \(a = OA = 1\), \(b=OB = 1\) and \(\theta=\angle AOB = 100^{\circ}\), we get \(A=\frac{1}{2}\times1\times1\times\sin100^{\circ}=\frac{1}{2}\sin100^{\circ}\).

Answer:

A. \(\frac{1}{2}\sin100^{\circ}\)