1. The Law of Sines! Homework: Lesson 12.3/1-10, 12-14, 19, 20
Objective: Be able to use the Law of Sines to find unknowns in triangles!
Homework:
Lesson 12.3/1-10, 12-14, 19, 20
Quiz Friday – 12.1 – 12.3

2. What if it’s not a right triangle? GASP!! What do we do then??
Quick Review:
What does Soh-Cah-Toa stand for?
What kind of triangles do we use this for?
right triangles
What if it’s not a right triangle? GASP!! What do we do then??

3. The Law of Sines: A B C a b c Note:
capital letters always stand for __________!
lower-case letters always stand for ________!
Use the Law of Sines ONLY when:
you DON’T have a right triangle AND
you know an angle and its opposite side
angles
sides

4. The Law of SINES
For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles:
Can also be written in this form:

5. AAS - 2 angles and 1 adjacent side
Use Law of SINES when ...
If 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 info to solve the Law of Sines equation.
Use the Law of Sines if you are given:
AAS - 2 angles and 1 adjacent side
ASA - 2 angles and their included side

6. Let’s do some problems! 
Ex. 1: Use the Law of Sines to find each missing angle or side. Round any decimal answers to the nearest tenth. A
63° C a 42 29
79˚
38˚

7. Ex. 2: Use the Law of Sines to find each missing angle or side
Ex. 2: Use the Law of Sines to find each missing angle or side. Round any decimal answers to the nearest tenth. s
40° T r
4.8
89°
51˚

8. Ex. 3: Draw ΔABC and mark it with the given information
Ex. 3: Draw ΔABC and mark it with the given information. Solve the triangle. Round any decimal answers to the nearest tenth. A B C a. c
76˚ 7
37˚
67˚ b

9. b. A B C
3.1
96˚ 12
70˚
14˚ b

10. Ex. 3: Draw ΔABC and mark it with the given information
Ex. 3: Draw ΔABC and mark it with the given information. Solve the triangle. Round any decimal answers to the nearest tenth.
You are given a triangle, ABC, with angle A = 70°, angle B = 80° and side a = 12 cm. Find the measures of angle C and sides b and c.
* In this section, angles are named with capital letters and the side opposite an angle is named with the same lower case letter .*

11. Ex. 3: con’t
The angles in a ∆ total 180°, so angle C = 30°.
Set up the Law of Sines to find side b: A C B
70°
80°
a = 12 c b

12. Ex. 3: con’t
Set up the Law of Sines to find side c: A C B
70°
80°
a = 12 c
b = 12.6
30°

13. Ex. 3: Solution
Angle C = 30°
Side b = 12.6 cm
Side c = 6.4 cm A C B
70°
80°
a = 12
c = 6.4
b = 12.6
30°
Note:
We used the given values of A and a in both calculations. Your answer is more accurate if you do not used rounded values in calculations.

14. Ex. 4: Draw ΔABC and mark it with the given information
Ex. 4: Draw ΔABC and mark it with the given information. Solve the triangle. Round any decimal answers to the nearest tenth.
You are given a triangle, ABC, with angle C = 115°, angle B = 30° and side a = 30 cm. Find the measures of angle A and sides b and c. A C B
115°
30°
a = 30 c b

15. We MUST find angle A first because the only side given is side a.
Ex. 4: con’t
To solve for the missing sides or angles, we must have an angle and opposite side to set up the first equation.
We MUST find angle A first because the only side given is side a.
The angles in a ∆ total 180°, so angle A = 35°. A C B
115°
30°
a = 30 c b

16. Set up the Law of Sines to find side b:
Ex. 4: con’t
Set up the Law of Sines to find side b: A C B
115°
30°
a = 30 c b
35°

17. Set up the Law of Sines to find side c:
Ex. 4: con’t
Set up the Law of Sines to find side c: A C B
115°
30°
a = 30 c
b = 26.2
35°

18. Angle A = 35° Side b = 26.2 cm Side c = 47.4 cm Ex. 4: Solution A C B
115°
30°
a = 30
c = 47.4
b = 26.2
35°
Note: Use the Law of Sines whenever you are given 2 angles and one side!

19. The Law of Sines
AAS
ASA
Use the Law of Sines to find the missing dimensions of a triangle when given any combination of these dimensions.

20. Example 3:
A 46-foot telephone pole tilted at an angle of from the vertical casts a shadow on the ground. Find the length of the shadow to the nearest foot when the angle of elevation to the sun is
Draw a diagram Draw Then find the

21. Example 3:
Since you know the measures of two angles of the triangle, and the length of a side opposite one of the angles you can use the Law of Sines to find the length of the shadow.

22. Example 3: Law of Sines Cross products Divide each side by sin
Use a calculator.
Answer: The length of the shadow is about 75.9 feet.

23. Your Turn:
A 5-foot fishing pole is anchored to the edge of a dock. If the distance from the foot of the pole to the point where the fishing line meets the water is 45 feet, about how much fishing line that is cast out is above the surface of the water?
Answer: About 42 feet of the fishing line that is cast out is above the surface of the water.

24. Example: Finding the Height of a Telephone Pole
15ft
15º
65º B A C

25. The Area of a Triangle Using Trigonometry
We can find the area of a triangle if we are given any two sides of a triangle and the measure of the included angle.
(SAS)

26. Area of an Oblique triangle
Using two sides and an Angle.