9.2 The Cosine Rule - SPM Additional Mathematics

9.2 The Cosine Rule

The cosine rule can be used when

(i) two sides and the included angle, or

(ii) three sides of a triangle are given.

(A) If you know 2 sides and 1 angle between them [included angle] ⇒ Cosine rule

Example:

Calculate the length of AC, x, in cm for the triangle above.

Solution:

\begin{array}{l} b^{2} = a^{2} + c^{2} - 2 a c \cos B \\ x^{2} = 4^{2} + 7^{2} - 2 (4) (7) \cos 50^{\circ} \\ x^{2} = 16 + 49 - 56 (0.6428) \\ x^{2} = 65 - 35.997 \\ x^{2} = 29.003 \\ x = 5.385 cm \end{array}

(B) If you know 3 sides ⇒ Cosine rule

Example:

Calculate ÐBAC for the triangle above.

Solution:

\begin{array}{l} \cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c} \\ \cos ∠ B A C = \frac{7^{2} + 6^{2} - 8^{2}}{2 (7) (6)} \\ \cos ∠ B A C = 0.25 \\ ∠ B A C = \cos^{- 1} 0.25 \\ ∠ B A C = {75.52}^{o} \end{array}