B.Math. 2005

Question

Let $0 < \theta < \pi$.

The area of the triangle in the plane formed by the vertices $(-1,0), (1,0), (\cos\theta, \sin\theta)$ is 

1. not more than $1$
2. can be more than $1$ but not more than $2$
3. can be more than $2$ but not more than $\pi$
4. can be more than $\pi$ but not more than $2\pi$

Comments

According to me graphically answer seems like option 2 ( can be more than 1 but not more than 2) but not sure.

Answer 1

Two coordinates of triangle lies on the x-axis, $(-1,0)$ and $(1,0)$.

so base of the triangle $= 2 $

height of triangle will be $=\sin \theta$

so area will be $\frac{1}{2} \times 2 \times \sin \theta = \sin \theta$

for $0<\theta<\pi$, maximum value of $\sin \theta$ will be $1$ only at $\theta = 90^{\circ}$.

Area of given triangle can be maximum $1$