• For any triangle \(ABC\):

\(a^2=b^2+c^2-2ab\cos{A}\)

\(b^2=a^2+c^2-2ab\cos{B}\)

\(c^2=a^2+b^2-2ab\cos{C}\)

• Where \(a\), \(b\), and \(c\) are the sides of the triangle, and \(A\), \(B\), and \(C\) are the angles opposite those sides.

• When two sides and the included angle are known.
• When all three sides are known, and you want to find an angle.

• Right-angled triangles, where the Pythagorean theorem and basic trigonometric ratios can be used instead.

• When two sides and the included angle are known, use the cosine rule to find the unknown side.

• When all three sides are known, use the cosine rule to find one of the angles.

• The cosine rule is a generalized form of the Pythagorean theorem that works for all types of triangles, not just right-angled ones.

In the diagram above, \(ABC\) is a scalene triangle.

Find the length of the \(AC\).

From diagram above, given:

\(AB=25\) cm,
\(BC=23\) cm, 
\(\angle{B}=40^\circ\).

Apply the cosine rule to find the length of \(AC\):

\(\begin{aligned} AC^2&=AB^2+BC^2-2(AB)(BC) \cos 40^\circ \\\\ &=25^2+23^2-2(25)(23) \cos 40^\circ \\\\ &=273.05. \\\\ AC&=16.52 \text{ cm}. \end{aligned}\)

• For any triangle \(ABC\):

\(a^2=b^2+c^2-2ab\cos{A}\)

\(b^2=a^2+c^2-2ab\cos{B}\)

\(c^2=a^2+b^2-2ab\cos{C}\)

• Where \(a\), \(b\), and \(c\) are the sides of the triangle, and \(A\), \(B\), and \(C\) are the angles opposite those sides.

• When two sides and the included angle are known.
• When all three sides are known, and you want to find an angle.

• Right-angled triangles, where the Pythagorean theorem and basic trigonometric ratios can be used instead.

• When two sides and the included angle are known, use the cosine rule to find the unknown side.

• When all three sides are known, use the cosine rule to find one of the angles.

• The cosine rule is a generalized form of the Pythagorean theorem that works for all types of triangles, not just right-angled ones.

In the diagram above, \(ABC\) is a scalene triangle.

Find the length of the \(AC\).

From diagram above, given:

\(AB=25\) cm,
\(BC=23\) cm, 
\(\angle{B}=40^\circ\).

Apply the cosine rule to find the length of \(AC\):

\(\begin{aligned} AC^2&=AB^2+BC^2-2(AB)(BC) \cos 40^\circ \\\\ &=25^2+23^2-2(25)(23) \cos 40^\circ \\\\ &=273.05. \\\\ AC&=16.52 \text{ cm}. \end{aligned}\)