The Trigonometric Functions
The Law of Cosines
Definition
• Let triangle ABC be any triangle with a, b, c
representing the measures of sides opposite angles
with measurements A, B, C respectively. Then the
following is true.
a 2 = b 2 + c 2 − 2bc cos A
b 2 = a 2 + c 2 − 2ac cos B
c 2 = a 2 + b 2 − 2ab cos C
Think about this situation
• Suppose that a roof is built so that the angle
at the peak and the lengths of the sides,
which differ, are known. How would the
width of the house be determined? Can you
use the Law of Sines?
This is why we have the Law of
Cosines
• Sometimes a situation can not be solved
using the Law of Sines. Sometimes you
need to use both the Law of Sines and the
Law of Cosines to solve a problem.
Example #1
• For a right handed golfer, a slice is a shot that curves to the right of its
intended path, and a hook curves off the the left. Suppose Mrs. Rodrigues
hits the ball from the seventh tee and the shot is a 160 yard slice 4 degrees
from the path straight to the cup. If the tee is 177 yards from the cup, how
far does the ball lie from the cup?
x 2 = 177 2 + 160 2 − 2(177)(160) cos 4 o
x 2 ≈ 426.9721933
Cup
x ≈ 20.66330548
X
Ball
177 yd
4
tee
160 yd
The ball is about 20.7 yds
way from the cup.
Example #2
• Solve a =19, b =24.3 c =21.8 Use the Law of Cosines
(24.3) 2 = (19 ) 2 + (21 .8) 2 − 2(19)(21 .8) cos B
590.49 = 361 + 475.24 − 828.4 cos B
590.49 = 836.24 − 828.4 cos B
− 245.75 = −828.4 cos B
.29666 = cos B
B ≈ 72.74 o
• Now use the Law of Sines
19
o
=
24.3
sin A
sin 72.74o
A ≈ 48.29
Now angle C = 58.97 degrees
Example # 3
• If you know the measures of three sides of a triangle, you can
find the area of the triangle by using the Law of Cosines and
1
the formula
K = bc sin A
2
• Find the area of triangle ABC if a =24, b =52 and
c =39. First solve for A by using the Law of Cosines
Now find the area
2
2
2
24 = 52 + 39 − 2(52)(39) cos A
576 = 2704 + 1521 − 4056 cos A
.89965 = cos A
A ≈ 25.89 o
1
(52)(39) sin 25.89 o
2
K = 442.7 units 2
K=
Hero’s Formula – A faster way!
• If the measure of the sides of a triangle are a, b, c,
then the area, K, of the triangle is found as follows.
K = s ( s − a )( s − b)( s − c)
1
where s = (a + b + c)
2
HW # 39
• Section 5.8
• Pp. 331-332
• # 11-21 odd, 25, 29