4100.21 – Find a Sine

Find a formula for sin⁡A\sin A, in terms of bb and cc only. (N.B. it's an isosceles triangle)

Solution

sin⁡A=hc\sin A = \dfrac{h}{c}

h=c2−(b2)2=c2−b24=4c2−b24=4c2−b22\begin{align*}h &= \sqrt{{c^2}-\left(\dfrac{b}{2}\right)^2} \\[1em]&= \sqrt{{c^2}-\dfrac{{b^2}}{4}} \\[1em]&= \sqrt{\dfrac{4c^2-b^2}{4}} \\[1em]&= \dfrac{\sqrt{4c^2-b^2}}{2}\end{align*}

So sin⁡A=4c2−b22c\sin A = \dfrac{\sqrt{4c^2-b^2}}{2c}.