# Trigonometry » Law of sines

Note: The length a and the angle size α (alpha) are used both, look closely at the difference between the two letters.

## Contents

What is the law of sines?
How do you calculate with the law of sines?
Proof of the law of sines

## What is the law of sines?

The law of sines is:
In every triangle ABC the following applies: a`sin`(α) = b`sin`(β) = c`sin`(γ)

This is true for a triangle ABC in which: α = A, β = B, γ = C.
And also a = BC, b = AC and c = AB.
This means that you can also write the law of sines as: BC`sin`(A) = AC`sin`(B) = AB`sin`(C)

## How do you calculate with the law of sines?

The easiest way is using a ratio table.
You fill in the table like this:

 a b c `sin`(α) `sin`(β) `sin`(γ)

Then you use cross multiplication to calculate the unknown sides and/or angles.

Example

Given is the following triangle.
Calculate the unknown angles and side.

Fill in the table:

 7.3 9.1 c `sin`(40°) `sin`(β) `sin`(γ)

First calculate β.
`sin`(β) = 9.1 × `sin`(40°)7.3 = 0.801...
β = `sin`–1(0.801...) = 53.253...°

As the sum of the angles in a triangle is 180°, you can calculate the third angle.
γ = 180° – α – β = 180° – 40 – 53.253...° = 86.747...°

With this new data, the table will look like this:

 7.3 9.1 c `sin`(40°) `sin`(53.253...°) `sin`(86.747...°)

Now we can calculate side c:
c = 7.3 × `sin`(86.747...°)`sin`(40°) ≈ 11.3

## Proof of the law of sines

To prove the law of sines, we use an altitude.

The following is now true:
`sin`(α) = hb  and  `sin`(β) = ha

Rearranging gives:
h = b · `sin`(α)  and  h = a · `sin`(β)

This gives:
a · `sin`(β) = b · `sin`(α)

Rearranging gives:
a`sin`(α) = b`sin`(β)

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