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 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
SSA (this is an ambiguous case)

4. Example 1
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 .*

5. Example 1 (con’t) A C B 70° 80° a = 12 c b
The angles in a ∆ total 180°, so angle C = 30°.
Set up the Law of Sines to find side b:

6. Example 1 (con’t) A C B 70° 80° a = 12 c b = 12.6 30°
Set up the Law of Sines to find side c:

7. Example 1 (solution) A C 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.

8. Example 2
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.

9. Example 2 (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

10. Example 2 (con’t) A C B 115° a = 30 c b
30°
a = 30 c b
35°
Set up the Law of Sines to find side b:

11. Example 2 (con’t) A C B 115° a = 30 c b = 26.2
30°
a = 30 c
b = 26.2
35°
Set up the Law of Sines to find side c:

12. Example 2 (solution) Angle A = 35° Side b = 26.2 cm Side c = 47.4 cm A
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!

13. The Ambiguous Case (SSA)
When given SSA (two sides and an angle that is NOT the included angle) , the situation is ambiguous. The dimensions may not form a triangle, or there may be 1 or 2 triangles with the given dimensions. We first go through a series of tests to determine how many (if any) solutions exist.

14. The Ambiguous Case (SSA)
In the following examples, the given angle will always be angle A and the given sides will be sides a and b. If you are given a different set of variables, feel free to change them to simulate the steps provided here.
‘a’ - we don’t know what angle C is so we can’t draw side ‘a’ in the right position A
B ? b
C = ?
c = ?

15. The Ambiguous Case (SSA)
Situation I: Angle A is obtuse
If angle A is obtuse there are TWO possibilities
If a ≤ b, then a is too short to reach side c - a triangle with these dimensions is impossible.
If a > b, then there is ONE triangle with these dimensions. A
B ? a b
C = ?
c = ? A
B ? a b
C = ?
c = ?

16. The Ambiguous Case (SSA)
Situation I: Angle A is obtuse - EXAMPLE
Given a triangle with angle A = 120°, side a = 22 cm and side b = 15 cm, find the other dimensions.
Since a > b, these dimensions are possible. To find the missing dimensions, use the Law of Sines: A B
a = 22
15 = b C c
120°

17. The Ambiguous Case (SSA)
Situation I: Angle A is obtuse - EXAMPLE
Angle C = 180° - 120° ° = 23.8°
Use Law of Sines to find side c: A B
a = 22
15 = b C c
120°
36.2°
Solution: angle B = 36.2°, angle C = 23.8°, side c = 10.3 cm

18. The Ambiguous Case (SSA)
Situation II: Angle A is acute
If angle A is acute there are SEVERAL possibilities.
Side ‘a’ may or may not be long enough to reach side ‘c’. We calculate the height of the altitude from angle C to side c to compare it with side a. A
B ? b
C = ?
c = ? a

19. The Ambiguous Case (SSA)
Situation II: Angle A is acute
First, use SOH-CAH-TOA to find h: A
B ? b
C = ?
c = ? a h
Then, compare ‘h’ to sides a and b . . .

20. The Ambiguous Case (SSA)
Situation II: Angle A is acute
If a < h, then NO triangle exists with these dimensions. A
B ? b
C = ?
c = ? a h

21. The Ambiguous Case (SSA)
Situation II: Angle A is acute
If h < a < b, then TWO triangles exist with these dimensions. A B b C c a h A B b C c a h
If we open side ‘a’ to the outside of h, angle B is acute.
If we open side ‘a’ to the inside of h, angle B is obtuse.

22. The Ambiguous Case (SSA)
Situation II: Angle A is acute
If h < b < a, then ONE triangle exists with these dimensions.
Since side a is greater than side b, side a cannot open to the inside of h, it can only open to the outside, so there is only 1 triangle possible! A B b C c a h

23. The Ambiguous Case (SSA)
Situation II: Angle A is acute
If h = a, then ONE triangle exists with these dimensions. A B b C c
a = h
If a = h, then angle B must be a right angle and there is only one possible triangle with these dimensions.

24. The Ambiguous Case (SSA)
Situation II: Angle A is acute - EXAMPLE 1
Given a triangle with angle A = 40°, side a = 12 cm and side b = 15 cm, find the other dimensions.
Find the height: A
B ?
15 = b
C = ?
c = ?
a = 12 h
40°
Since a > h, but a< b, there are 2 solutions and we must find BOTH.

25. The Ambiguous Case (SSA)
Situation II: Angle A is acute - EXAMPLE 1
FIRST SOLUTION: Angle B is acute - this is the solution you get when you use the Law of Sines! A B
15 = b C c
a = 12 h
40°

26. The Ambiguous Case (SSA)
Situation II: Angle A is acute - EXAMPLE 1
SECOND SOLUTION: Angle B is obtuse - use the first solution to find this solution.
In the second set of possible dimensions, angle B is obtuse, because side ‘a’ is the same in both solutions, the acute solution for angle B & the obtuse solution for angle B are supplementary.
Angle B = ° = 126.5°
a = 12 A B
15 = b C c
40°
1st ‘a’
1st ‘B’

27. The Ambiguous Case (SSA)
Situation II: Angle A is acute - EXAMPLE 1
SECOND SOLUTION: Angle B is obtuse
Angle B = 126.5°
Angle C = 180°- 40° ° = 13.5°
a = 12 A B
15 = b C c
40°
126.5°

28. The Ambiguous Case (SSA)
Situation II: Angle A is acute - EX. 1 (Summary)
Angle B = 53.5°
Angle C = 86.5°
Side c = 18.6
Angle B = 126.5°
Angle C = 13.5°
Side c = 4.4
a = 12 A B
15 = b C
c = 4.4
40°
126.5°
13.5° A B
15 = b C
c = 18.6
a = 12
40°
53.5°
86.5°

29. The Ambiguous Case (SSA)
Situation II: Angle A is acute - EXAMPLE 2
Given a triangle with angle A = 40°, side a = 12 cm and side b = 10 cm, find the other dimensions. A
B ?
10 = b
C = ?
c = ?
a = 12 h
40°
Since a > b, and h is less than a, we know this triangle has just ONE possible solution - side ‘a’opens to the outside of h.

30. The Ambiguous Case (SSA)
Situation II: Angle A is acute - EXAMPLE 2
Using the Law of Sines will give us the ONE possible solution: A B
10 = b C c
a = 12
40°

31. The Ambiguous Case - Summary
if angle A is acute
find the height,
h = b*sinA
if angle A is obtuse
if a < b  no solution
if a > b  one solution
if a < h  no solution
if h < a < b  2 solutions
one with angle B acute,
one with angle B obtuse
if a > b > h  1 solution
If a = h  1 solution
angle B is right
(Ex I)
(Ex II-1)
(Ex II-2)

32. The Law of Sines AAS ASA SSA (the ambiguous case)
Use the Law of Sines to find the missing dimensions of a triangle when given any combination of these dimensions.

33. Additional Resources Web Links: