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: asin(α) = bsin(β) = csin(γ)

Triangle with letters at the correct spot, see text below

This is true for a triangle ABC in which: α = angle signA, β = angle signB, γ = angle signC.
And also a = BC, b = AC and c = AB.
This means that you can also write the law of sines as: BCsin(angle signA) = ACsin(angle signB) = ABsin(angle signC)

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.
Triangle with a=9.1 b=7.3 and α=40°

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.
triangle ABC with the altitude from C drawn called h

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:
asin(α) = bsin(β)


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