Lesson 2: The Law of Cosines
Estimated time: 30-35 minutes
Learning Objectives
- State the Law of Cosines in all three forms
- Solve SAS triangles (two sides and included angle)
- Solve SSS triangles (all three sides given)
- Choose between Law of Sines and Law of Cosines appropriately
The Law of Cosines
Law of Cosines:
c² = a² + b² − 2ab cosC
a² = b² + c² − 2bc cosA
b² = a² + c² − 2ac cosB
This generalizes the Pythagorean theorem (when C=90°, cosC=0 and c²=a²+b²).
SAS: Two Sides and Included Angle
Example 1: SAS
a=8, b=11, C=40°. Find c.
c² = 64 + 121 − 2(8)(11)cos40° = 185 − 176(0.766) = 185 − 134.8 = 50.2
c = √50.2 ≈ 7.09
Then use Law of Sines to find angles A and B.
SSS: Three Sides Given
Example 2: SSS
a=5, b=7, c=10. Find angle C (the largest angle, opposite the longest side).
cosC = (a²+b²−c²)/(2ab) = (25+49−100)/(70) = −26/70 = −0.3714
C = arccos(−0.3714) ≈ 111.8°
Example 3: Choosing the Right Law
Use Law of Sines when you know: AAS, ASA, or SSA.
Use Law of Cosines when you know: SAS or SSS.
Applications
Example 4: Navigation
Two ships leave a port. Ship A travels 100 mi on bearing N 30° E, Ship B travels 80 mi on bearing S 50° E. Find the distance between them.
The angle at the port between the two paths = 30° + 50° = 80° (they are not supplementary because the bearings are on the same side). Actually: the angle between N30E and S50E at the port is 180° − 30° − 50° = 100°.
c² = 100² + 80² − 2(100)(80)cos100° = 10000+6400−16000(−0.1736) = 16400+2778 = 19178
c ≈ 138.5 miles
Check Your Understanding
1. a=6, b=9, C=75°. Find c.
2. a=3, b=5, c=7. Find angle C.
3. When is cosC negative, and what does that tell you?
Key Takeaways
- Law of Cosines: c² = a² + b² − 2ab cosC.
- Use for SAS (find the third side) and SSS (find an angle).
- It generalizes the Pythagorean theorem to non-right triangles.
- Always find the largest angle first when given SSS.