Investigation: The Cosine Law
Completion requirements
| Investigation |
The Cosine Law
Open the Cosine Law applet to begin exploring this relationship.
In the applet, you are given two expressions for the triangle:
-
\(a^2\)
- \(b^2 + c^2 - 2bc\cos A\)
Move the points around to make various triangles. Compare the two expressions. What do you notice about their values, and does that agree with the cosine law?
Key Lesson Marker |
The Cosine Law
The cosine law used to solve for a side of a non-right triangle is
\(\begin{align}
a^2 &= b^2 + c^2 - 2bc\cos A \\
b^2 &= a^2 + c^2 - 2ac\cos B \\
c^2 &= a^2 + b^2 - 2ab\cos C \\
\end{align}\)
a^2 &= b^2 + c^2 - 2bc\cos A \\
b^2 &= a^2 + c^2 - 2ac\cos B \\
c^2 &= a^2 + b^2 - 2ab\cos C \\
\end{align}\)
The cosine law used to solve for an angle in a non-right triangle is
\(\begin{align}
\cos A &= \frac{{b^2 + c^2 - a^2 }}{{2bc}} \\
\cos B &= \frac{{a^2 + c^2 - b^2 }}{{2ac}} \\
\cos C &= \frac{{a^2 + b^2 - c^2 }}{{2ab}} \\
\end{align}\)
The cosine law is useful in two circumstances.
- Determining a side when you are given the other two sides and the angle between the known sides (Figure 1).
- Determining an angle when you are given all three sides of a triangle (Figure 2).
The best way to understand how the cosine law is used is to do some examples.