1. Sec. 7.4: The Laws of Cosine and Sine
Objectives:
Use trig-values to determine the measure of a missing side or angle of a non-right triangle.
Determine the area of a triangle given side–ange–side information.

2. Concept: Parts of a Triangle
The Law of Cosines is all about the relationship of the given information within a non-right triangle.
Triangles are named by their vertices (angles) which use all upper-case letters. B
Sides are named by using the same letter of the opposite angle, but in lower case. c a A C b

3. Concept: Law of Consines
The Law of Cosines is used when we have the lengths of all 3 sides (SSS) and need an angle, or
When we have an angle and the two sides that make it (SAS) and need the third side.

4. Concept: Law of Consines cont…
The Law of Cosines can be written:
a2 = b2 + c2 – (2bc)CosA or
b2 = a2 + c2 – (2ac)CosB
c2 = a2 + b2 – (2ab)CosC
Remaining 2 sides
Opposing side and angle: 1 given, 1 variable

5. Concept: Using the Law of Consines
Using the given measurements, solve for the missing part of the triangle: Q 8
80º
q = ______
mLP = ______
mLR = ______ 6 P q R
Solving for side q:
Q and q are opposing parts and mLQ is given.
1. Write the formula q2 = p2 + r2 – (2pr)CosQ
2. Substitute q2 = – (2)(8)(6)Cos80º

6. Concept: Using the Law of Consines
Using the given measurements, solve for the missing part of the triangle: Q 8
80º
q = ______
mLP = ______
mLR = ______ 6 P q R
Solving for side q:
Since there are no variables on the right side you can just enter this in your calculator the way it is written.
3. Enter into calculator q2 = – 96Cos80º

7. Concept: Using the Law of Consines
Using the given measurements, solve for the missing part of the triangle: Q 8
80º
q = ______
mLP = ______
mLR = ______
9.13 6 P q R
9.13
Solving for side q:
Simplify the right side q2 = …
5. Take the square root 2nd x nd (–) =
√q2 = √ …
q = 9.13

8. Concept: Using the Law of Consines
Using the given measurements, solve for the missing part of the triangle: Q 8
80º
q = ______
mLP = ______
mLR = ______
9.13 6 P q R
=9.13
Solving for LP
1. Set up the Equation p2 = q2 + r2 – (2qr)CosP
2. Substitute 62 = – (2)(9.13)(8)CosP
3. Simplify 36 = – CosP
4. Combine like terms 36 = – CosP

9. Concept: Using the Law of Consines
Using the given measurements, solve for the missing part of the triangle: Q 8
80º
q = ______
mLP = ______
mLR = ______
9.13 6 P q R
=9.13
Solving for LP
36 = – CosP
Move the constant – –
– = –146.08CosP

10. Concept: Using the Law of Consines
Using the given measurements, solve for the missing part of the triangle: Q 8
80º
q = ______
mLP = ______
mLR = ______
9.13 6 P q R
=9.13
Solving for LP
– = –146.08CosP
________ ___________
6. Divide BS by Coef – –146.08
0.7623…= CosP

11. Concept: Using the Law of Consines
Using the given measurements, solve for the missing part of the triangle: Q 8
80º
q = ______
mLP = ______
mLR = ______
9.13 6
40.3º
40.3º P q R
=9.13
Solving for LP
0.7623…= CosP
7. Undo cos 2nd cos 2nd (–) ent
40.3º = mLP

12. Concept: Using the Law of Consines
Using the given measurements, solve for the missing part of the triangle: Q 8
80º
q = ______
mLP = ______
mLR = ______
9.13 6
40.3º
40.3º P q R
=9.13
Solving for LR
Now that you have the measure of 2 angles you can use the Triangle Sum Postulate to find the measure of the missing angle.
80º º + mLR = 180º
120.3º + mLR = 180º
mLR = 59.7º

13. Concept: Law of Sines
The relationship between the angles of a triangle and the opposite sides can be written as a proportion.
Sine of the angle /
length of opposite side/
You MUST have values for two opposing parts of the triangle. You need the measure of an angle and the side opposite that angle.

14. Concept: Law of Sines cont…B
Find the length of side b.
85º
8cm
mLA = 40º. The side across from it = 8cm.
40º C A b
1. Set up the proportion.
Sin 40º Sine 85
_______ _______ = b

15. Concept: Law of Sines cont…B
Find the length of side b.
85º
8cm
2. Cross multiply.
40º C A b
Sin 40º Sine 85
_______ _______ = b
8sin85º
= bsin40º

16. Concept: Law of Sines cont…B
Find the length of side b.
85º
8cm
Divide both sides by
sin40º.
40º C A b
8sin85º
= bsin40º
_______ _______
sin40º sin40º
8sin85º = b
sin40º

17. Concept: Law of Sines cont…B
Find the length of side b.
85º
8cm
4. Put into calculator
40º C A b
8sin85º = b
sin40º
8 Sin ( ) ENT ÷ sin ( ) ENT
b = cm

18. Concept: Area Area of a non-right triangle: You MUST have SASB
Area of a non-right triangle:
You MUST have SAS
85º
6cm
Area = bcsinA 2
35º C A
8cm
Area = (6)(8)sin35º 2
Area =13.77cm2