The Law

1
The Law
of SINES
2
The Law of SINES
For any triangle (right, acute or obtuse), you
may use the following formula to solve for
missing sides or angles
3
Use Law of SINES when ...
you have 3 dimensions of a triangle and you need
to find the other 3 dimensions - they cannot be
just ANY 3 dimensions though, or you wont have
enough info to solve the Law of Sines equation.
Use the Law of Sines if you are given

• AAS - 2 angles and 1 adjacent side
• ASA - 2 angles and their included side
• SSA (this is an ambiguous case)

4
Example 1

• You are given a triangle, ABC, with angle A
  70, angle B 80 and side a 12 cm. Find the
  measures of angle C and sides b and c.
• In this section, angles are named with capital
  letters and the side opposite an angle is named
  with the same lower case letter .

5
Example 1 (cont)
The angles in a ? total 180, so angle C
30. Set up the Law of Sines to find side b
6
Example 1 (cont)
Set up the Law of Sines to find side c
7
Example 1 (solution)
Angle C 30 Side b 12.6 cm Side c 6.4 cm
Note We used the given values of A and a in both
calculations. Your answer is more accurate if you
do not used rounded values in calculations.
8
Example 2

• You are given a triangle, ABC, with angle C
  115, angle B 30 and side a 30 cm. Find the
  measures of angle A and sides b and c.

9
Example 2 (cont)
To solve for the missing sides or angles, we must
have an angle and opposite side to set up the
first equation. We MUST find angle A first
because the only side given is side a. The
angles in a ? total 180, so angle A 35.
10
Example 2 (cont)
Set up the Law of Sines to find side b
11
Example 2 (cont)
Set up the Law of Sines to find side c
12
Example 2 (solution)
Angle A 35 Side b 26.2 cm Side c 47.4 cm
Note Use the Law of Sines whenever you are
given 2 angles and one side!
13
The Ambiguous Case (SSA)

• When given SSA (two sides and an angle that is
  NOT the included angle) , the situation is
  ambiguous. The dimensions may not form a
  triangle, or there may be 1 or 2 triangles with
  the given dimensions. We first go through a
  series of tests to determine how many (if any)
  solutions exist.

14
The Ambiguous Case (SSA)
In the following examples, the given angle will
always be angle A and the given sides will be
sides a and b. If you are given a different set
of variables, feel free to change them to
simulate the steps provided here.
15
The Ambiguous Case (SSA)
Situation I Angle A is obtuse If angle A is
obtuse there are TWO possibilities
If a b, then a is too short to reach side c - a
triangle with these dimensions is impossible.
If a gt b, then there is ONE triangle with these
dimensions.
16
The Ambiguous Case (SSA)
Situation I Angle A is obtuse - EXAMPLE
Given a triangle with angle A 120, side a 22
cm and side b 15 cm, find the other
dimensions.
Since a gt b, these dimensions are possible. To
find the missing dimensions, use the Law of Sines
17
The Ambiguous Case (SSA)
Situation I Angle A is obtuse - EXAMPLE
Angle C 180 - 120 - 36.2 23.8 Use Law of
Sines to find side c
Solution angle B 36.2, angle C 23.8, side
c 10.3 cm
18
The Ambiguous Case (SSA)
Situation II Angle A is acute If angle A is
acute there are SEVERAL possibilities.
Side a may or may not be long enough to reach
side c. We calculate the height of the
altitude from angle C to side c to compare it
with side a.
19
The Ambiguous Case (SSA)
Situation II Angle A is acute
First, use SOH-CAH-TOA to find h
Then, compare h to sides a and b . . .
20
The Ambiguous Case (SSA)
Situation II Angle A is acute
If a lt h, then NO triangle exists with these
dimensions.
21
The Ambiguous Case (SSA)
Situation II Angle A is acute
If h lt a lt b, then TWO triangles exist with these
dimensions.
If we open side a to the outside of h, angle B
is acute.
If we open side a to the inside of h, angle B
is obtuse.
22
The Ambiguous Case (SSA)
Situation II Angle A is acute
If h lt b lt a, then ONE triangle exists with these
dimensions.
Since side a is greater than side b, side a
cannot open to the inside of h, it can only open
to the outside, so there is only 1 triangle
possible!
23
The Ambiguous Case (SSA)
Situation II Angle A is acute
If h a, then ONE triangle exists with these
dimensions.
If a h, then angle B must be a right angle and
there is only one possible triangle with these
dimensions.
24
The Ambiguous Case (SSA)
Situation II Angle A is acute - EXAMPLE 1
Given a triangle with angle A 40, side a 12
cm and side b 15 cm, find the other
dimensions.
Find the height
Since a gt h, but alt b, there are 2 solutions and
we must find BOTH.
25
The Ambiguous Case (SSA)
Situation II Angle A is acute - EXAMPLE 1
FIRST SOLUTION Angle B is acute - this is the
solution you get when you use the Law of Sines!
26
The Ambiguous Case (SSA)
Situation II Angle A is acute - EXAMPLE 1
SECOND SOLUTION Angle B is obtuse - use the
first solution to find this solution.
In the second set of possible dimensions, angle B
is obtuse, because side a is the same in both
solutions, the acute solution for angle B the
obtuse solution for angle B are
supplementary. Angle B 180 - 53.5 126.5
27
The Ambiguous Case (SSA)
Situation II Angle A is acute - EXAMPLE 1
SECOND SOLUTION Angle B is obtuse
28
The Ambiguous Case (SSA)
Situation II Angle A is acute - EX. 1 (Summary)
Angle B 126.5 Angle C 13.5 Side c 4.4
Angle B 53.5 Angle C 86.5 Side c 18.6
29
The Ambiguous Case (SSA)
Situation II Angle A is acute - EXAMPLE 2
Given a triangle with angle A 40, side a 12
cm and side b 10 cm, find the other
dimensions.
Since a gt b, and h is less than a, we know this
triangle has just ONE possible solution - side
aopens to the outside of h.
30
The Ambiguous Case (SSA)
Situation II Angle A is acute - EXAMPLE 2
Using the Law of Sines will give us the ONE
possible solution
31
The Ambiguous Case - Summary
32
The Law of Sines
33
Additional Resources

• Web Links

• http//oakroadsystems.com/twt/solving.htmSineLaw
• http//oakroadsystems.com/twt/solving.htmDetectiv
  e