# Extended Sine Law

Sine law is one of the easiest topics in trigonometry. It’s my job to make it more complicated for you to be challenged.

Given a $\Delta ABC$ and Circle $O$ above. The following relation is called extended sine law.

$2R=\displaystyle\frac{a}{sinA}=\displaystyle\frac{b}{sinB}=\displaystyle\frac{c}{sinC}$

It’s the same as the sine law but we added another figure, a circumscribing circle. Now, it’s time practice!

Worked Problem 1:

In $\Delta ABC$, $\angle{B}=45$°, $\angle{C}=60$°. How long is side $\overline{AB}$   if   $\overline{AC}=3$.

Solution:

Again, this is trigonometry so let’s draw.

Based on the image above, it’s easy to recognize that the unknown can be solved using sine law.

By sine law we have,

$\displaystyle\frac{\overline{AB}}{sin60}=\displaystyle\frac{3}{sin45}$

$\overline{AB}=\displaystyle\frac{3}{sin45}\cdot sin60$

$\overline{AB}=\displaystyle\frac{3}{\frac{\sqrt{2}}{2}}\cdot \displaystyle\frac{\sqrt{3}}{2}$

$\overline{AB}=\displaystyle\frac{3\sqrt{6}}{2}$

Worked Problem 2:

If a circle circumscribed the given triangle above, what is the area of the circle?

Solution:

By sine law,

$2R=\displaystyle\frac{3}{sin45}$

$R=\displaystyle\frac{3}{2sin45}$

$R=\displaystyle\frac{3\sqrt{2}}{2}$

Solving for the area of a circle,

$A=\pi R^2$

$A=\pi (\displaystyle\frac{3\sqrt{2}}{2})^2$

$A=\displaystyle\frac{9\pi}{2}$  sq. units

Worked Problem 3:

A regular pentagon is inscribed in a circle with radius of 10 cm. Find the length of each side of pentagon.

Solution:

Solving for $\theta$:

$5\theta=360$

$\theta=72$°

In $\Delta EHD$,

$\gamma+\gamma +\theta =180$

$2\gamma+72=180$

$2\gamma=108$

$\gamma=54$°

Using Sine Law, Let s be the length of the side of octagon.

$\displaystyle\frac{s}{sin\theta}=\displaystyle\frac{r}{sin\gamma}$

$\displaystyle\frac{s}{sin72}=\displaystyle\frac{10}{sin54}$

$s=\displaystyle\frac{10sin72}{sin54}$

$s=11.76cm$

I just have 1 quick question, without using any search engine. Who formulated sine law?