Sine Rule

Only one of these formulas should be used at a time.

Note;

A, B and C represent angles while a, b and c represent sides. This indicates that the capital letters must be considered as angles while the small letters as sides.

Examples;

1. In a triangle ABC, A = 54⁰, B = 67⁰, a = 13.9m. Find b and c.

Solution

Step 1

From Sine Rule;

Substitution;

Step 2

Cross Multiply;

b X Sin 54⁰ = 13.9m X Sin 67⁰

b X 0.8090 = 13.9m X 0.9205

0.8090b = 12.79495

Step 3

Divide both sides by 0.8090;

Step 4

To find c, we will need to find angle C;

Angle C = 180⁰ – (54⁰ + 67⁰)

Angle C = 180⁰ - 121⁰

C = 59⁰.

Step 5

Substitution;

Step 6

Cross Multiply;

0.9205 X c = 0.8572 X 15.8m

0.9205c = 13.54376

Step 7

Divide both sides by 0.9205;

2. In a triangle PQR, R = 53⁰, q = 3.6m, r = 4.3m. Find Q.

Solution

Step 1

From Sine Rule;

Step 2

Cross Multiply;

4.3m X Sin Q = 3.6m X 0.7986

4.3mSinQ = 2.8751

Step 3

Divide both sides by 4.3m;

Sin Q = 0.6686

Practice these questions below;

1. In a triangle ABC, if A = 38⁰, B = 27⁰, b = 17m. Find a and c.

2. In a triangle XYZ, if X = 69⁰, y = 9cm, x = 19cm. Find Y and z.

3. In a triangle PQR, if Q = 36⁰, P = 88⁰, p = 9.5cm. Find q and r.

4. In the triangle EFG below, find G;

5. In a triangle ABC, if A = 39⁰, a = 8.2m, b = 5.6m. Find B and c.