# 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?

### Dan

Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.