Download 9C The Sine Rule

9C The Sine Rule
Similar to the cosine rule, the sine rule is used to find an angle or a side in a nonright angled triangle. The difference is it works when you have:
 One side and any two angles
 Two sides and an angle opposite one of the sides
Remember, the upper case letters are the angles, and
the lower case letters are the sides (opposite the angle
with the same letter).
a
b
c


(use this version when finding a side length)
sin A sin B sinC
sin A sin B sinC


(use this version when finding an angle)
a
b
c
Example 1
Find the value of x.
The Ambiguous Case
This arises when you are using the sine rule to find an angle, and it simply means
there may be zero, one or two possible triangles with the given measurements.
Essentially one of three things will happen when you attempt to find an angle
using the sine rule:
1. You get sin A = 1 which means there is one possible triangle (it is a
right-triangle).
2. You get sin A = a number bigger than one which means no triangle can
be drawn with the given measurements.
3. You get sin A = a number between 0 and 1 which means there may be
one or two triangles with the given measurements. Next find the
supplement of angle A.
a. If (supplement angle A + given angle) < 180° then two triangles
are possible.
b. If (supplement angle A + given angle) > 180° then only one
triangle is possible.
(The additional handout explains in more detail than you really need all the
possible scenarios.)
Example 2
Find the measure of angle C in the triangle below.
sin C sin B

c
b
sin C sin 25

11
7
sin 25
sin C 
 11
7
 sin 25

C  sin 1 
 11
 7

 41.6
Check for the ambiguous case by finding the supplement of angle C, C ’ = 138.4°
Since (138.4° + 25°) < 180° there are two possible values for angle C.
C = 41.6° or 138.4°
Example 3
Is it possible to have a triangle with the measurements shown? Explain.