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. A C B
70°
80°
a = 12 c b

6. 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.

7. A C B
115°
30°
a = 30 c b

8. 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.

9. 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 = ?

10. The Ambiguous Case (SSA)
Situation I: Angle A is obtuse
If angle A is obtuse there are TWO possibilities
If a < b, NO triangle can be formed
If a > b, then there is ONE triangle with these dimensions. A
B ? a b
C = ?
c = ? A
B ? a b
C = ?
c = ?

11. 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.

12. The Ambiguous Case (SSA)
Situation I: Angle A is obtuse - EXAMPLE
Given a triangle with angle A = 93°, side a = 12 cm and side b = 13 cm, find the other dimensions.

13. 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

14. 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 . . .

15. 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

16. 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.

17. 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

18. 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.

19. 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. A
B ?
15 = b
C = ?
c = ?
a = 12 h
40°

20. 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°

21. 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)

22. 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.

23. Page 303 (9 – 23 odd) ================ Pages 303-304 (25 – 45 odd)