Question

If $(\cos \alpha ,\sin \alpha ,0)(\cos \beta ,\sin \beta ,0),(\cos \gamma ,\sin \gamma ,0)$ are vertices of a triangle the circum radius $R$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A $1$

Look the three vertices of the triangle

They all three satisfy that ${\cos }^2\alpha +{\sin }^2\alpha =1$

${\cos }^2\beta +{\sin }^2\beta =1$

${\cos }^2\gamma +{\sin }^2\gamma =1$

Thus they all three lie on circle $x^2+y^2=1$

Or you can see that for the circle $x^2+y^2=1\Rightarrow \left(\cos \theta ,\sin \theta \right)$ is the general point on the circle which is satisfied by all three vertices of given triangle for $\theta =\alpha ,\beta ,\gamma$. Thus the triangle lies on circle

$x^2+y^2=1$