1. The Law
of SINES

2. The Law of SINES
For any triangle (right, acute or obtuse), you may use the following formula to solve for missing sides or angles:

3. Use Law of SINES when ... AAS - 2 angles and 1 adjacent side
you have 3 dimensions of any triangle and you need to find the one or more of the other dimensions
Use the Law of Sines if you are given:
AAS - 2 angles and 1 adjacent side
ASA - 2 angles and their included side
SSA- 2 sides and a obtuse angle opposite them

4. #1. You are given a triangle, ABC, with angle A = 70°, angle B = 80° and side a = 12 cm. Find the measure of side b . A C B
70°
80°
a = 12 c b

5. #2, Find c Fill length of side b Find angle C
Set up the Law of Sines to find side c: B
80°
a = 12 c
70°
30° A C
b = 12.6

6. We now know all the side lengths and angle sizes of the triangleC B
70°
80°
a = 12
c = 6.4
b = 12.6
30°
Angle C = 30°
Side b = 12.6 cm
Side c = 6.4 cm
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.

7. We MUST find angle A first because the only side given is side a.
#3. 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
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°.

8. #3 continued Set up the Law of Sines to find side b: 30° 35° A C B
115°
30°
a = 30 c b
35°
Set up the Law of Sines to find side b:

9. #4. Find side c Set up the Law of Sines to find side c: 30° 35° A C B
115°
30°
a = 30 c
b = 26.2
35°
Set up the Law of Sines to find side c:

10. Example 2 (solution) Angle A = 35° Side b = 26.2 cm Side c = 47.4 cm
115°
30°
a = 30
c = 47.4
b = 26.2
35°
Angle A = 35°
Side b = 26.2 cm
Side c = 47.4 cm
Note: Use the Law of Sines whenever you are given 2 angles and one side!

11. C
SOLUTION: 10 12 A
180 B
= A

12. Solving an SAS Triangle
The Law of Sines is good for
ASA - two angles and the included side
AAS - two angles and any side
SSA - two sides and an opposite angle (being aware of possible ambiguity)
Why would the Law of Sines not work for an SAS triangle?
26° 15
12.5
No side opposite from any angle to get the ratio

13. Deriving the Law of Cosines
Write an equation using Pythagorean theorem for shaded triangle.
b h a
k c - k
A B c

14. Law of Cosines
Visualize the Law of Sines and Law of Cosines
Geogebra

15. Law of Cosines
Law of Cosines can be used to solve a non-right triangle when you know one angle and two sides
Note the pattern

16. Applying the Cosine Law
Now use it to solve the triangle we started with
Label sides and angles
Side c first C 15
26°
12.5
A B c

17. Applying the Cosine Law
Now calculate the angles
use and solve for B C 15
26°
12.5
A B
c = 6.65

18. Applying the Cosine Law
The remaining angle determined by subtraction
180 – – 26 = 60.25 C 15
26°
12.5
A B
c = 6.65

19. Wing Span
The leading edge of each wing of the B-2 Stealth Bomber measures feet in length. The angle between the wing's leading edges is °. What is the wing span (the distance from A to C)?
Hint … use the law of cosines! C A

20. Using the Cosine Law to Find Area
Recall that
We can use the value for h to determine the area C
b h a
A B c

21. Using the Cosine Law to Find Area
We can find the area knowing two sides and the included angle
Note the pattern C
b a
A B c

22. Try It Out Determine the area of these triangles 42.8° 127° 76.3° 17.924 12
76.3°

23. Cost of a Lot
An industrial piece of real estate is priced at $4.15 per square foot. Find, to the nearest $1000, the cost of a triangular lot measuring 324 feet by 516 feet by 412 feet.
324
412
516