The Law of Sines or Law of Cosines Notes
When using the Law of Cosines it is necessary to know the angle included between two sides (SAS) or all
three sides of the triangle (SSS). However, there are times the angle that is known is not the angle
included between two known sides. And there are other times where we might know two angles and only
one side (AAS or ASA). In both cases the Law of Cosines cannot be used. In those cases you can use the
Law of Sines.
NOTE If you HAVE
and you
WANT the missing
then you should use:
SAS
leg
__________________
SAS or SSS
angles
__________________
AAS or ASA
angle
__________________
AAS or ASA
leg
__________________
SSA
leg(s)
__________________
SSA
angle(s)
__________________
Procedure for using Law of Cosines or Law of Sines
1. Draw a picture of the triangle given. Label the sides and angles given.
2. Check for the triangle congruencies given (AAS, SSA, ASA, SAS, SSS)
3. From this, choose to use the Law of Sines or Law of Cosines. If using the Law of Sines, determine the
number of triangles possible. (See Step 4). If using Law of Cosines, move to Step 6.
4. Check for the triangle congruencies given (AAS, SSA, ASA)
a. If given AAS or ASA, subtract from 180o to find the missing angle and use the Law of Sines to find
the missing sides, move to Step 6.
b. If given SSA, determine the number of triangles possible (Step 5)
5. Ambiguous case of the Law of Sines
Angle A
Triangle
Measurements
Acute
a < b sin A
Acute
a = b sin A
Acute
b sin A < a < b
Acute
a>b
Obtuse or Right
a>b
Obtuse or Right
a<b
# of triangles
formed
0
1
2
1
1
0
a. If 1 triangle is possible, continue to Step 6
b. If 2 triangles are possible, redraw triangles if necessary, find a missing angle using Law of Sines
and subtract from 180o to find an angle in the other possible triangle. Continue to Step 6.
6. Find the missing sides and angles using appropriate law.
Formulas
Law of Sines
sin 𝐴 sin 𝐵 sin 𝐶
=
=
𝑎
𝑏
𝑐
Law of Cosines
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
𝐴 = cos
−1
𝑏 2 + 𝑐 2 − 𝑎2
(
)
2𝑏𝑐
𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵
𝐵 = cos
−1
𝑎2 + 𝑐 2 − 𝑏 2
(
)
2𝑎𝑐
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶
𝐶 = cos
−1
𝑎2 + 𝑏 2 − 𝑐 2
(
)
2𝑎𝑏
First determine whether the Law of Sines or the Law of Cosines should be used first to solve each
triangle described below. Then sketch and solve each of the above triangles on a separate sheet of
paper.
1. A = 40º, b= 6, c = 7
_______________________________
2. a = 14, b = 15, c = 16
_______________________________
3. a = 10, A = 45º , c = 8
_______________________________
4. A = 40º, C = 80º, c = 14
_______________________________
5. b = 17 , B = 42.5º, a = 11
_______________________________
6. C = 35º, a =11, b = 10.5
_______________________________
Determine whether the Law of Sines or Law of Cosines should be used to solve each triangle described
below. Then solve each of the triangles on a separate sheet of paper.
1.
∠𝐴 = 25𝑜 , 𝑎 = 125, 𝑏 = 150
2.
3.
A boat captain spies a buoy 32o to the left of the direction of travel. After the boat travels 2 miles, the
buoy is 54o to the left side. How far is the buoy from the boat at the time of the second sighting?
4
On a baseball diamond with 90 foot sides, the pitcher’s mound is 60.5 feet from home plate. How far
is the pitcher’s mound from third base?
5.
What is the smallest side of a triangle that has angles 42o and 20o with the side between them
measuring 56 cm?
6.
A ship is 75 miles from one radio transmitter and 140 miles from another. If the angle between the
signals is 120o, how far apart at the transmitters?
7.
Solve the following triangles, if possible. If there is a second possible triangle, solve it as well.
a.
∠𝐴 = 65𝑜 , b = 7, a = 10
b.
∠𝐴 = 41𝑜 , a = 9, b = 12
c.
∠𝐴 = 33𝑜 , b = 19, a = 12
∠𝐴 = 118𝑜 , 𝑎 = 17, 𝑏 = 11