Question

A triangle ABC is inscribed within the unit circle. Let x be the measure of the angle C. Express the length of AB in terms of x.

Answer 1

1. Check the picture attached.

2. m(AOB)=2x as it is a central angle, intercepting the same arc (AB) as ACB.

3. AO=OB=1, as they are radii of the unit circle.

4. From the cosine theorem:

$AB^{2} =AC^{2} + BC^{2} -2*AC*BC*cos C$

substituting the known values:

$AB^{2} =1^{2} + 1^{2} -2*1*1*cos x$

$AB^{2} =2 -2*cos x=2(1-cosx)$

$AB= \sqrt{2(1-cosx)}$