# Law of Sines – Explanation & Examples

To this point, our love affair with Trigonometry has been restricted to right triangles. But how do we determine the unknown parts (sides and angles) if we’re dealing with oblique triangles? Through the law of sines.

The Law of Sines simply relates the lengths of the legs of any triangle to the sines of its corresponding angles. Using the law of sines, we get the flexibility to solve the oblique triangles.

This lesson aims to clear up any confusion you might have about the concepts involving the Law of Sines. We will be also able to answer the following questions.

• What is the law of sines?
• When to use the law of sines?
• How can we use the law of sines?

## What Is the Law of Sines?

There is a special relationship which we can determine the missing parts of oblique triangles, and this relationship is known as the Law of Sines.

Consider a triangle $△ABC$ with the sides $a$, $b$, and $c$ facing the angles $\alpha$, $\beta$, and $\gamma$, respectively, as shown in Figure 13-1.

Note that the side ${\displaystyle {\overline {AC}}}$ of the triangle is along the positive direction of the x-axis with vertex $A$ at the origin, so $∠BAC$ is in the standard position. Thus,

${\displaystyle {\overline {AB}}} = c$

$m∠BAC = \alpha$

Therefore, the coordinates of the point $B$ will be $(c \cos \alpha, c \sin \alpha)%%EDITORCONTENT%%nbsp; Let us shift the origin from$A$to$C$, as shown in Figure 13-2. Looking at the above diagram,$∠BCX$is in the standard position now. Thus,$BC = am∠BCX = 180^{\circ } – \gamma$Therefore, the coordinates of$B$are$[a \cos (180^{\circ } – \gamma), a \sin (180^{\circ } – \gamma)]$As the y-coordinate of B remains the same in both cases.$a \sin (180^{\circ } – \gamma) = c \sin \alphaa \sin \gamma = c \sin \alpha\frac{a}{\sin\:\alpha\:}=\:\frac{c}{\sin\:\gamma}%%EDITORCONTENT%%nbsp;    — [1]

Similarly, if the side $AB$ is along the positive x-axis, we can determine that: