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.

sinelaw1

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.

sinelaw2

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:

sinelaw3

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:

sinelaw4

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

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.
Dan

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