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= √(2(1-cosx))`