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Section 13.2 Law of Cosines
The Law of Cosines is useful for relating the sides of triangles that are not right triangles.
You will recall that if we had a right triangle we defined the cosine as
adjacent
but this requires that we have a right angle so that we can find the
cos(θ ) =
Hypotenuse
length of they hypotenuse.
Law of Cosines:
C
b
a
A
B
c
a = b + c − 2bc(cos A)
2
2
2
OR
b2 + c2 − a2
cos A =
2bc
b = a + c − 2ac(cos B )
a2 + c2 − b2
cos B =
2ac
c = a + b − 2ab(cos C )
b2 + a2 − c2
cos C =
2ab
2
2
2
2
2
2
The law of cosines is useful when you know:
1) Three sides
2) Two sides and the included angle.
Recall:
1. The sum of the angles in a triangle is 180 degrees so if you know any two angles
you know the third. A + B + C = 180
2. The height of a triangle is less than or equal to the length of two of the sides.
Ex 1: Use the information to solve the triangle.
a=3
b=8
C
A
B
c=9
Step 1: Find the largest angle first. That will always be the angle opposite the longest
side. In this case that would be angle C
8 2 + 32 − 9 2
b2 + a2 − c2
The formula we use is cos C =
so cos C =
2ab
2(3)(8)
cos C = −.16667
and so C = 99.59°
Step 2: Use law of sines to solve for the other angles.
a
b
c
=
=
sin A sin B sin C
3
8
9
=
=
sin A sin B sin 99.59
You will need your calculator and some algebra to get:
sin A = 0.32867
so A = 19.19°
sin B = 0.87647
so B = 61.22°
.
Ex 2: A = 55°, b = 3, c = 10.
Step 1: Draw a triangle and label the sides with what you know..
a
C
b=3
55
B
c=10
Step 2: use the law of cosines to find a.
a 2 = b 2 + c 2 − 2bc(cos A)
a 2 = 3 2 + 10 2 − 2(3)(10)(cos 55)
a = 8.64
Use the law of sines to find the other two angles.
sin A sin B
=
a
b
And
sin 55 sin B
=
8.64
3
B = 16.52
Then C = 180 –B – A = 108.48
sin A sin C
=
a
c
sin 55 sin C
=
8.64
10
Ex 3: A triangular parcel of land has 115 meters of water front and the other boundaries
have lengths of 76 meters and 92 meters. What angles does the water front make with the
other two boundaries?
Ex 4: On a baseball diamond with 90 – foot sides, the pitcher’s mound is 60.5 feet from
home plate. How far is it from the pitcher’s mound to third base?