Law of Cosines
Generalised Pythagorean theorem for any triangle.
Key idea
Generalised Pythagorean theorem for any triangle.
In depth
c² = a² + b² − 2ab cos C. Use it when you know two sides and the included angle (SAS) or all three sides (SSS). It reduces to Pythagoras when C = 90°.
Worked example
Sides 7, 9, 12: cos C = (49 + 81 − 144)/(2·7·9) = −14/126 ≈ −0.111, C ≈ 96.4°.
Formula
Real-world uses
- Calculating the distance between two ships from their distances to a lighthouse and the angle
Common mistake
Dropping the 2 or the negative sign.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Law of Cosines?
Generalised Pythagorean theorem for any triangle. It is a topic at the advanced level in Trigonometry.
What is the formula for Law of Cosines?
The key formula is c² = a²+b²−2ab cosC.
Where is Law of Cosines used in real life?
Real-world examples: Calculating the distance between two ships from their distances to a lighthouse and the angle.
What should I learn before Law of Cosines?
Make sure you are comfortable with Ambiguous Case of Law of Sines, the previous topic in Trigonometry.
What is a common mistake with Law of Cosines?
Dropping the 2 or the negative sign.
How can I practise Law of Cosines?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Law of Cosines?
Move on to Area of Triangle with Sine next.
Related advanced topics in Trigonometry
- Graph of Tangent – Periodic with vertical asymptotes at odd multiples of π/2.
- Phase Shift – Horizontal translation of a wave.
- Vertical Shift and Midline – Moves the wave up or down.
- General Sinusoid – Combine amplitude, period, phase and shift.
- Graphs of Reciprocal Trig Functions – Secant, cosecant and cotangent graphs.
- Pythagorean Identity – The key identity from the unit circle.