BeginnerTopic 16 of 104Engineering Mathematics
Law of sines and cosines
Solve any triangle; used in surveying, truss geometry and force resolution.
In depth
The law of sines a/sinA=b/sinB=c/sinC and the law of cosines c²=a²+b²-2ab cosC solve any triangle, not just right ones. They are used in triangulation, force polygons and finding resultants of non-perpendicular vectors.
Key points
- Use cosines for SAS or SSS
- Use sines for ASA or AAS
- The ambiguous SSA case may give two triangles
ExampleTwo forces 30 N and 40 N at 60° have resultant √(900+1600+1200)≈ 60.8 N.
What is Law of sines and cosines?
Solve any triangle; used in surveying, truss geometry and force resolution.
Why is it important to understand law of sines and cosines?
Triangles are the building blocks of trusses and surveys, and these laws let you find every side and angle from partial information.
When do I use the law of sines instead of the law of cosines?
Use sines when you know a side and its opposite angle, plus one more value. Use cosines when you know two sides and the included angle, or all three sides.
Can the law of sines give two answers?
Yes, in the ambiguous case where the given angle is opposite the shorter side. Check whether both triangles are geometrically valid.
How is Law of sines and cosines used in practice?
Typical applications include finding a member force in a triangular truss, computing the distance across a river from two bearings and solving the geometry of a four-bar linkage.
What should I learn before Law of sines and cosines?
Study "Unit circle and identities" first (topic 15 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Law of sines and cosines?
Use cosines for SAS or SSS. Use sines for ASA or AAS. The ambiguous SSA case may give two triangles.
Can you give an example of Law of sines and cosines?
Two forces 30 N and 40 N at 60° have resultant √(900+1600+1200)≈ 60.8 N.
What comes after Law of sines and cosines?
The next topic is "Coordinate geometry", which builds directly on this one.
How difficult is Law of sines and cosines?
It is classed as Beginner in Engineering Mathematics (topic 16 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.