8'2 The Law of Cosines more solving triangles SAS and SSS

1
8.2 The Law of Cosines more solving
trianglesSAS and SSS
Mrs. Kessler
2
The Law of Cosines

• If A, B, and C are the angles of a triangle, and
  a, b, and c are the lengths of the sides opposite
  these angles, then
• a2 b2 c2 - 2bc cos A
• b2 a2 c2 - 2ac cos B
• c2 a2 b2 - 2ab cos C.
• The square of a side of a triangle equals the sum
  of the squares of the other two sides minus twice
  their product times the cosine of their included
  angle.

3
Solving an SAS Triangle

• Use the Law of Cosines to find the side opposite
  the given angle.
• Use the Law of Sines to find the angle opposite
  the shorter of the two given sides. This angle is
  always acute.
• Find the third angle. Subtract the measure of the
  given angle and the angle found in step 2 from
  180º.

4
Example 1

• Solve the triangle shown with A 60º, b 20,
  and c 30.

a b 20 c 30 A 60º B
C
Use the Law of Cosines to find the side opposite
the given angle, a.
a2 b2 c2 - 2bc cos A
a2 202 302 - 2(20)(30) cos 60º
a2 400 900 - 1200(0.5) 700
a 26.5
5
Example 1 cont.
Solve the triangle shown with A 60º, b 20,
and c 30.
a 26.5 b 20 c 30 A 60º B
C
Step 2 Use the Law of Sines to find the angle
opposite the shorter of the two given sides.
This angle is always acute. The shorter of the
two given sides is b 20. Thus, we will find
acute angle B.
6
Example 1 cont.
Solve the triangle shown with A 60º, b 20,
and c 30.
a 26.5 b 20 c 30 A 60º B
41º C
Step 3 Find the third angle.
C 180º - A - B ? 180º - 60º - 41º 79º
The solution is a ? 26.5, B ? 41º, and C ? 79º.
7
Solving an SSS Triangle

• Use the Law of Cosines to find the angle opposite
  the longest side.
• Use the Law of Sines to find either of the two
  remaining acute angles.
• Find the third angle. Subtract the measures of
  the angles found in steps 1 and 2 from 180º.

8
Example 2
a 6 b 9 c 4 A B
C
Solve triangle ABC if a 6, b 9, and c 4
1. Use the Law of Cosines to find the angle
opposite the longest side.
b2 a2 c2 - 2ac cos B
C
92 62 42 - 2(6)(4) cos B
B
A
9
Example 2 cont'd
a 6 b 9 c 4 A B
127.2? C
Solve triangle ABC if a 6, b 9, and c 4

• 2. Use the Law of Sines to find either of the two
  remaining acute angles.

C
B
A
10
Example 2 cont'd
a 6 b 9 c 4 A 32.1? B
127.2? C
C
B
A
11
Ex. 3 Try thisSolve triangle ABC if a 10,
b 3, and C 15º
a b c A
B C
12
Ex 3. cont'd Solve triangle ABC if a 10, b
3, C 15º
a b c A
B C
B 6º, A 159º c 7
13
Ex 4. Solve triangle ABC if a 4, b 7, and c
6
a b c A
B C
14
Ex 4. cont'd Solve triangle ABC if a 4, b
7, and c 6
B 86º, A 35º C 59º
15
Area of An Oblique Triangle

• The area of a triangle equals one-half the
  product of the lengths of two sides times the
  sine of their included angle. In the following
  figure, this wording can be expressed by the
  formulas

16
Example 1

• Find the area of a triangle having two sides of
  lengths 24 meters and 10 meters and an included
  angle of 62º.

17
Example 2

• Find the area of a triangle having two sides of
  lengths 12 ft. and 20 ft. and an included angle
  of 57º.

18
Example 3 Find the area of the following
triangle

• A 56º, C 24º, a 12

Draw the triangle
Now what?
C
Use the Law of Sines to get the needed side.
B 100º
c 5.9 b 14.3
B
A
19
Example 3 Find the area of the following
triangle

• A 56º, C 24º, a 12

Which formula?
Draw the triangle
C
b 14.3
B
c 5.9
A
20
OR this way.. Find the area of the
following triangle

• A 56º, C 24º, a 12

c 5.9 b 14.3 B100?
Which formula?
Draw the triangle
C
a 12
b 14.3
B
c 5.8
A
21
Area of a triangle with no angles known
Know from 7.1 Area
22
Area of a triangle with no angles known
23
Herons Formula

• The area of a triangle with sides a, b, and c is

24
Example

• Use Herons formula to find the area of the given
  trianglea10m, b 8m, c 4m

Solution
25
Example

• Two airplanes leave an airport at the same time
  on different runways. One flies at a bearing of
  N66ºW at 325 miles per hour. The other airplane
  flies at a bearing of S26ºW at 300 miles per
  hour. How far apart will the airplanes be after
  two hours?

Solution After two hours. the plane flying at
325 miles per hour travels 325 2 miles, or 650
miles. Similarly, the plane flying at 300 miles
per hour travels 600 miles. The situation is
illustrated in the figure. Let b the distance
between the planes after two hours. We can use a
north-south line to find angle B in triangle
ABC. Thus, B 180º - 66º - 26º 88º. We now
have a 650, c 600, and B 88º.
26
Example cont.
Two airplanes leave an airport at the same time
on different runways. One flies at a bearing of
N66ºW at 325 miles per hour. The other airplane
flies at a bearing of S26ºW at 300 miles per
hour. How far apart will the airplanes be after
two hours?
b2 6502 6002 - 2(650)(600) cos 88º
Substitute a 650, c 600, and B 88.
? 755,278 Use a calculator.
After two hours, the planes are approximately
869 miles apart.
27
let's program

• 17, 16, 14 are sides of a triangle. What is the
  area?

28
The Law of Cosines