TGT - Law of Sines Lesson

Law of Sines

This module is all about solving triangles that are oblique. One of the ways that we do it is by using the Law of Sines. Below we will discover the Law of Sines.

Consider $\bigtriangleup ABC$ below.

Draw an altitude from ∠B, down to side b and call it h.

Now, we could set up the following sine ratios using side h.

$\sin A=\frac{h}{c}\:and\:\sin C=\frac{h}{a}\\ csinA=h\:\:\:\:\:\:\:asin C=h$

Using the substitution, we can say that:

$c\:sinA=a\:\sin C\\ \frac{c}{\sin C}=\frac{a}{\sin A}$

Well, what if we drew an altitude from ∠A, down to side a and call it j.

Now, we could set up the following sine ratios using side j.

$\sin B=\frac{j}{c}\:and\:\sin C=\frac{j}{b}\\ sin B=j\:\:\:\:\:\:\:b\:\sin C=j$

Using the substitution, we can say that:

$c\sin B=b\sin C\\ \frac{c}{\sin C}=\frac{b}{\sin B}$

And now, we know that:

$\frac{c}{\sin C}=\frac{b}{\sin B}\:and\:\frac{c}{\sin C}=\frac{a}{\sin A}$  

So, by the transitive property we know:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Therefore, LAW OF SINES states: Let a, b, and c be the side lengths opposite angles A, B, and C. Then $\bf\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$.

Watch this video to practice solving a right triangle using the Law of Sines:

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