Transcript The Laws of SINES

The Law
of
COSINES
The Law of COSINES
For any triangle (right, acute or obtuse), you may
use the following formula to solve for missing
sides or angles:
2
2
2
2
2
2
a  b  c  2bccos A
b  a  c  2accos B
c  a  b  2abcosC
2
2
2
Use Law of COSINES 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 won’t have enough
information to solve the Law of Cosines equations. Use
the Law of Cosines if you are given:


SAS - 2 sides and their included angle
SSS
Example 1: Given SAS
Find all the missing dimensions of triangle ABC, given
that angle B = 98°, side a = 13 and side c = 20.
Use the Law of Cosines equation that uses a, c and B to find
side b:
b 2  a 2  c 2  2accos B
B
98°
C
c = 20
b
b 2  132  202  2 13 20cos98
A
b 2  641.37
b  25.3
Example 1: Given SAS
Now that we know B and b, we can use the Law of
Sines to find one of the missing angles:
B
a = 13
98°
C
c = 20
b = 25.3
Solution:
b = 25.3, C = 51.5°, A = 30.5°
25.3
20

sin98 sin C
1 20sin 98
C  sin 

25.3
A
C  51.5
A  180  98  51.5  30.5
Example 2: Given SAS
Find all the missing dimensions of triangle, ABC,
given that angle A = 39°, side b = 20 and side c = 15.
Use the Law of Cosines equation that uses b, c and A to find
side a:
a 2  b 2  c 2  2bccos A
B
a
c = 15
A
39°
a 2  202  152  2  2015cos39
b = 20
C
a 2  158.71
a  12.6
Example 2: Given SAS
Use the Law of Sines to find one of the missing angles:
B
a = 12.6
c = 15
A
39°
b = 20
C
12.6
15

sin39 sin C
1 15sin39
C  sin 
12.6 
C  48.5
B  180  39  48.5  92.5
Important: Notice that we used the Law of Sine equation to find angle C
rather than angle B. The Law of Sine equation will never produce an obtuse
angle. If we had used the Law of Sine equation to find angle B we would
have gotten 87.5°, which is not correct, it is the reference angle for the
correct answer, 92.5°. If an angle might be obtuse, never use the Law of
Sine equation to find it.
Example 3: Given SSS
Find all the missing dimensions of triangle, ABC,
given that side a = 30, side b = 20 and side c = 15.
We can use any of the Law of
Cosine equations, filling in a, b
& c and solving for one angle.
A
c = 15
B
b = 20
a = 30
C
Once we have an angle, we
can either use another Law of
Cosine equation to find
another angle, or use the Law
of Sines to find another angle.
Example 3: Given SSS
Important: The Law of Sines will never produce an obtuse angle. If
an angle might be obtuse, never use the Law of Sines to find it. For
this reason, we will use the Law of Cosines to find the largest angle
first (in case it happens to be obtuse).
Angle A is largest because side a is largest:
302  202  152  2 2015cosA
A
900 400 225 600cosA
c = 15
B
a 2  b 2  c 2  2bccosA
275 600cosA
b = 20
a = 30
C
275
 cosA
600
1  275 
A  cos 
 117.3

600
Example 3: Given SSS
A
c = 15
B
117.3°
a = 30
Solution:
A = 117.3°
B = 36.3°
C = 26.4°
Use Law of Sines to find angle B or C
(its safe because they cannot be obtuse):
b = 20
C
30
20

sin117.3 sin B
1 20sin117.3
B  sin 

30
B  36.3
C  180  117.3  36.3  26.4
The Law of Cosines
2
2
2
2
2
2
a  b  c  2bccos A
b  a  c  2accos B
c 2  a 2  b 2  2abcosC
When given one of these dimension
combinations, use the Law of Cosines
to find one missing dimension and
then use Law of Sines to find the rest.

SAS
 SSS
Important: The Law of Sines will never produce an obtuse angle. If
an angle might be obtuse, never use the Law of Sines to find it.