Cosine Rule

The formulas for Cosine rule are;

a² = b² ­+ c² – 2bc Cos A    OR

b² = a² + c² – 2ac Cos B    OR

c² = a² + b² – 2ab Cos C

Note;

Where A, B and C represent angles and a, b and c represent sides.

Examples;

1.

Solution

Step 1

Identify the sides of each angle A, B and C. Take note, the side a particular angle is facing belong to it.

a = ycm, b = 3cm, c = 5cm and A = 102⁰.

From Cosine Rule;

a² = b² + c² – 2bc Cos A

Step 2

Substitution;

y² = 3² + 5² – 2(3) (5) Cos 102⁰

y² = 9 + 25 – 30 (- 0.2079)

y² = 34 + 6.2374

y² = 40.2374

2. 

Solution

Step 1

Identify the sides of each angle A, B and C. Take note, the side a particular angle is facing belongs to it.

Where a = 3cm, b = 2cm, c = xcm and C = 80⁰

From Cosine Rule;

c² = a² + b² – 2ab Cos C

Substitution;

x² = 3² + 2² – 2 (3) (2) Cos 80⁰

x² = 9 + 4 – 12 (0.1736)

x² = 13 – 12 X 0.1736

x² = 13 – 2.0832

x² = 10.1968

3. In a triangle ABC, a = 7.92m, c = 8.44m and B = 151.3⁰. Calculate |AC|.

Solution

Step 1

Draw a triangle representing all the given sides and angles. Take note, the side a particular angle is facing belongs to it.

a = 7.92m, b = ?, c = 8.44m and B = 151.3⁰.

From Cosine Rule;

b² = a² + c² -2ac Cos B

|AC|² = 7.92² + 8.44² – 2 (7.92) (8.44) Cos 151.3⁰

|AC|² = 62.7264 + 71.2336 – 133.6896 (-0.8771)

|AC|² = 133.96 + 117.2653

|AC|² = 251.2253

Practice these questions below;

Calculate the length of the side opposite the given angle in each of triangles ABC given below;

5. In a triangle ABC, if a = 7.2cm, A = 72⁰ and b = c. Find b.