The Cosine Rule (Leaving Cert Mathematics): Revision Notes

Worked Example 1: Finding a Missing Side

Problem: In triangle ABC, $|AB| = 6$, $|AC| = 8$, and $\angle BAC = 48°$. Find $|BC|$ to the nearest integer.

Solution: Let $|BC| = a$. Using the Cosine Rule formula $a^2 = b^2 + c^2 - 2bc \cos A$:

Step 1: Substitute the known values $a^2 = 8^2 + 6^2 - 2(8)(6) \cos 48°$

Step 2: Calculate each term $a^2 = 64 + 36 - 96 \cos 48°$ $a^2 = 100 - 96(0.6691)$ $a^2 = 100 - 64.23 = 35.77$

Step 3: Find the final answer $a = \sqrt{35.77} = 5.98$

Answer: $|BC| = 6$ (to the nearest integer)

Worked Example 2: Finding a Missing Angle

Problem: Find $\angle ABC$ in a triangle where the sides are $a = 6$, $b = 5$, and $c = 4$.

Solution: To find angle $B$, we rearrange the formula $b^2 = c^2 + a^2 - 2ca \cos B$:

Step 1: Substitute known values $5^2 = 4^2 + 6^2 - 2(4)(6) \cos B$ $25 = 16 + 36 - 48 \cos B$

Step 2: Solve for cosine B $25 = 52 - 48 \cos B$ $48 \cos B = 52 - 25 = 27$ $\cos B = \frac{27}{48}$

Step 3: Find the angle Using the inverse cosine function: $B = \cos^{-1}\left(\frac{27}{48}\right) = 55.77°$

Answer: $\angle ABC = 56°$ (to the nearest degree)

Worked Example 3: Step-by-Step Process

Problem: In triangle ABC with sides $a = 7$, $b = 11$, and included angle $C = 72°$, find side $c$.

Step 1: Identify what you know and what you need to find

• Known: two sides and included angle
• Unknown: third side
• Method: Cosine Rule

Step 2: Choose the correct form of the formula $c^2 = a^2 + b^2 - 2ab \cos C$

Step 3: Substitute the values $c^2 = 7^2 + 11^2 - 2(7)(11) \cos 72°$ $c^2 = 49 + 121 - 154 \cos 72°$ $c^2 = 170 - 154(0.309)$ $c^2 = 170 - 47.59 = 122.41$

Step 4: Solve for the unknown $c = \sqrt{122.41} = 11.1$