The Sine Law
A. The Sine Law
The sine law is a relationship that is true for all triangles, including ones that do not have a right angle. With the use of the sine law, unknown side lengths and angle measures of right and non-right triangles can be determined.
Open the Sine Law applet (8 February 2013, Created with GeoGebra) to begin exploring this relationship. Use the applet to answer the following questions.
Move the points to see how the ratios of the sine of an angle and the side length across from it are related.
- How are angle A and side a related? How about angle B and side b? Angle C and side c?
- How are the ratios
, 
, and 
related. Move the points around to see if this relationship holds for different triangles.
When labelling a triangle in this section, angles are labelled with capital letters and sides are labelled with lowercase letters. The angle and the side across it share the same letter name.
When using the Sine Law applet, you may have noticed that , , and were for any triangle created. This leads to three equations known as the sine law.
These three equations are sometimes written as the single expression
. This relationship can also be expressed as 
.
When using either of these relationship formats to solve for unknowns, you will only use two of the three ratios at a time (as shown in the bullets above).
In the following example you will see part of a proof of the sine law. You will complete the proof in Lesson 4.4 Game On!
Example 1
Prove that 
for any triangle that is labelled as shown.
| Proof | Explanation |
|---|---|
|
Draw a line perpendicular to c that passes through C. Label this line d.
|
This produces two right triangles. You will now be able to use the sine ratio. |
 and  |
The first equation comes from the triangle on the left. The second equation comes from the triangle on the right. |
|
b sin A = d and a sin B = d
|
The two equations have been rearranged to isolate d. |
|
b sin A = a sin B
|
b sin A and a sin B are both equal to d and so are equal to each other. This is an example of the transitive property of equality. |
|
|
The previous equation has been rearranged to the desired format. |
