Sine Rule (Law of Sines)
The law of sines is a relationship linking the sides of a triangle with the sine of their corresponding angles.
Sine Rule
Given the following triangle with corresponding side lengths and :
the sine rule or law of sines is the following identity:
We will prove the first identity
The second equality can be proved similarly.
By drawing the height of the triangle from vertex to the opposite side, we can express the height in two different ways:
- First, we have , which implies
- Also, , which implies
By equating these values of , we have
By drawing the height from the other two vertices, we can similarly show the second equality.
See the extended sine rule for another proof.
Examples
One real-life application of the sine rule is the sine bar, which is used to measure the angle of a tilt in engineering. Other common examples include measurement of distances in navigation and measurement of the distance between two stars in astronomy.
In the following triangle, suppose that and .
What is the side length
From the sine rule, we have
Therefore, .
Ambiguous Case
A common application of the sine rule is to determine the triangle given some of its sides and angles. The ambiguous case refers to scenarios where there are 2 distinct triangles that satisfy such a configuration. This occurs when we are given the angle-side-side, as shown in the diagram below:
If the side lengths of are and with opposite to measuring degrees, what is the measure of opposite to
By the sine rule, we have or . Solving for yields or .
However, note that . Since and another possible measure of is approximately .
Using the same example given above, find the measure of if the side lengths were swapped: and .
By the sine rule, we have or . Solving for yields or .
However, note that . Since but it is not true that there is only one possible measure of , which is approximately degrees.
Using the very first example given above, if you are further given the angle of to be 13 degrees, what is the total possible number of distinct measures of
By the sine rule we have or , which is clearly false, implying there is no such triangle. Hence there is no possible triangle that fits these criteria.
The sides of a triangle are and and the angles opposite to them are and respectively. Given and what is the number of triangles that can be formed from the given data?
Extended Sine Rule
The extended sine rule is a relationship linking the sides of a triangle with the sine of their corresponding angles and the radius of the circumscribed circle. The statement is as follows:
Given triangle , with corresponding side lengths and and as the radius of the circumcircle of triangle , we have the following:
Note: The statement without the third equality is often referred to as the sine rule. The relationship between the sine rule and the radius of the circumcircle of triangle is what extends this to the extended sine rule.
Extended Sine Rule
Let be the center of the circumcircle, and the midpoint of Then is perpendicular to . Now, observe that is equal to or where depending on whether is in the triangle or not. Then or , and thus . As such,
Show that the area of triangle is equal to
Let be the foot of the perpendicular from to . Using as the base and as the height, the area of the triangle is . From right triangle , . Thus, the area of the triangle is , which is often quoted. Now, using the extended sine rule, we have, , and thus the area of the triangle is
A spherical buoy is floating in the sea with an emerged cap part of height and radius as shown below:
What is the radius of this buoy?