1. The Laws of SINES
and COSINES

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 given ...
AAS
ASA
SSA (the 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 since your answer is more accurate if you do not use 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/angles, we must have an angle/side opposite pair to set up the first equation.
We MUST find angle A 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 those 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
b = 15 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
b = 15 C c
120°
36.2°
Solution: angle B = 36.2°, angle C = 23.8°, side c = 10.3

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)
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 a right angle
if angle A is acute
find the height,
h = b*sinA

25. The Law of COSINES

26. Use Law of COSINES when given ...
SAS
SSS (start with the largest angle!)