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# Law of Sines

The Law of Sines is one of the most useful triangle equations we can find in the world of math. If you've already tackled the Pythagorean theorem, you're already aware of how interesting triangles can be. The Law of Sines takes this one step further, allowing us to make all kinds of calculations based on the angles of triangles rather than their sides. But how do we use the Law of Sines? Let's find out.

## What is the Law of Sines?

Also known simply as the "Sine Rule," the Law of Sines is represented by the following equation:

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

Remember, the lower-case letters represent the sides of a triangle, whereas the upper-case letter represents the angles of a triangle. A is always opposite to a, B is always opposite to b, and C is always opposite to c.

But what about the phrase sin?

"Sin" represents "sine." Alone with cosine and tangent, these values represent the ratio of a side in a right-angled triangle. Sine is written like this:

$\mathrm{sin}\theta =\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$

$\theta$ "theta" represents the angle.

Remember, the hypotenuse is only found in right triangles but we can still take the sin of any angle regardless of whether we are talking about an acute, obtuse, or right triangle.

## Testing the Law Of Sines

But does the Law of Sines really make sense?

There's only one way to find out! Let's test the Sine Rule on a triangle:

Let's say this triangle has three angles:

$A=62.2°,B=33.5°,C=84.3°$

This triangle also has three sides:

$a=8$ , $b=5$ , and $c=9$

We can plug these values into the Sine Rule as follows:

$\frac{a}{\mathrm{sin}A}=\frac{8}{\mathrm{sin}\left(62.2\right)}=\frac{8}{0.885}=9.04$

$\frac{b}{\mathrm{sin}B}=\frac{5}{\mathrm{sin}\left(33.5\right)}=\frac{5}{0.552}=9.06$

$\frac{c}{\mathrm{sin}C}=\frac{9}{\mathrm{sin}\left(84.3\right)}=\frac{9}{0.995}=9.04$

As you can see, the answers are almost identical. As far as quick calculations go, this is good enough to suggest the Law of Sines is valid.

## How does the Law of Sines work?

Essentially, the Law of Sines works by turning non-right triangles (also known as "oblique triangles") into right triangles. Consider the following triangles:

As you can see, we can turn the first triangle into two triangles that share the same side: h. The same logic applies to the second triangle

Remember, the sine of an angle is the opposite divided by the hypotenuse. Based on this knowledge, we can create the following equations:

$\mathrm{sin}\left(A\right)=\frac{h}{b}$ or $b\mathrm{sin}\left(A\right)=h$

Also:

$\mathrm{sin}\left(B\right)=\frac{h}{a}$ or $a\mathrm{sin}\left(B\right)=h$

Because $\left(a\mathrm{sin}\left(B\right)\right)$ and $\left(b\mathrm{sin}\left(A\right)\right)$ are both equal to h, we are left with:

$a\mathrm{sin}\left(B\right)=b\mathrm{sin}\left(A\right)$

We can then rewrite this equation as:

$\frac{a}{\mathrm{sin}\left(A\right)}=\frac{b}{\mathrm{sin}\left(B\right)}$

## Using the Law of Sines to find unknown values

Now that we understand how the Law of Sines works, we can have some fun with it!

Using this formula, we can solve many unknown values on a triangle. One thing that we can do is find unknown sides.

For example, let's say we have a triangle with the following properties:

We only know the value of one side: $b=7$

However, we know two of its angles: $B=35°$ and $C=105°$ .

Remember, the Law of Sines states that:

$\frac{a}{\mathrm{sin}\left(A\right)}=\frac{b}{\mathrm{sin}\left(B\right)}=\frac{c}{\mathrm{sin}\left(C\right)}$

Let's plug in these values:

$\frac{a}{\mathrm{sin}A}=\frac{7}{\mathrm{sin}\left(35\right)}=\frac{c}{\mathrm{sin}\left(105\right)}$

We don't really need to know anything about a/sin A, so we can eliminate this from the equation.

$\frac{c}{\mathrm{sin}\left(105\right)}=\frac{7}{\mathrm{sin}\left(35\right)}$

$c=\frac{7}{\mathrm{sin}\left(35\right)}×\mathrm{sin}\left(105\right)$

$c=\frac{7}{0.5784}×0.966$

$c=11.8$

We can also use the Law of Sines to find unknown values.

But before we do this, let's turn a few fractions upside down. You'll soon see why this makes things a lot easier.

So instead of:

$\frac{a}{\mathrm{sin}A}$

We would write:

$\frac{\mathrm{sin}A}{a}$

This would leave us with the following equation:

$\frac{\mathrm{sin}\left(A\right)}{a}=\frac{\mathrm{sin}\left(B\right)}{b}=\frac{\mathrm{sin}\left(C\right)}{c}$

Now let's imagine we have a triangle with the following values:

We know that this triangle has the following sides: $c=5.5$ and $b=4.7$

We also have the value of a single angle: $C=63°$ .

$\frac{\mathrm{sin}\left(A\right)}{a}=\frac{\mathrm{sin}\left(B\right)}{b}=\frac{\mathrm{sin}\left(C\right)}{c}$

$\frac{\mathrm{sin}\left(A\right)}{a}=\frac{\mathrm{sin}\left(B\right)}{4.7}=\frac{\mathrm{sin}\left(63\right)}{5.5}$

We can focus on the last 2 parts of our chained equality.

$\frac{\mathrm{sin}\left(B\right)}{4.7}=\frac{\mathrm{sin}\left(63\right)}{5.5}$

$\mathrm{sin}\left(B\right)=\frac{\mathrm{sin}\left(63\right)}{5.5}×4.7$

$\mathrm{sin}\left(B\right)=0.7614$

$B={\mathrm{sin}}^{-1}\left(0.7614\right)=49.6°$

## Examples of the Law of Sines in action

Here are some more examples of the Law of Sines in action:

To find the third unknown angle:

$\frac{45}{\mathrm{sin}\left(30\right)}=\frac{b}{\mathrm{sin}\left(20\right)}=\frac{c}{\mathrm{sin}\left(130\right)}$

and

Example 2:

To find the third unknown angle: $180-42-75=63$

$\frac{a}{\mathrm{sin}\left(42\right)}=\frac{b}{\mathrm{sin}\left(75\right)}=\frac{22}{\mathrm{sin}\left(63\right)}$

$a=16.52$ and $b=23.85$

In some situations, there may be multiple "right answers."

If you have two sides and one known angle, there are three possibilities:

• The triangle does not exist.
• There are two triangles that may have these values
• There is only one triangle that has these values

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