Sine vs Cosine Rule: When to Use Each

 

Choosing the Right Trig Rule

When solving triangle problems, selecting the correct trigonometric rule is key. Use this decision guide to help you choose based on the information given in the question.

 

What You Know	What You Want	Rule to Use
Two sides and an angle opposite one side	An angle opposite the other side	Sine Rule
Two angles and a side opposite one angle	The other opposite side	Sine Rule
Two sides and the included angle	The third side	Cosine Rule
All three sides	Any angle	Cosine Rule
Two sides and the included angle	Area of the triangle	Area Rule ($\frac{1}{2}ab\sin C$)

 

 

 

 

 

Quick Checklist

Are you working with a right-angled triangle? Use SOHCAHTOA.

Do you have one or two angle-side pairs? Try the sine rule.

Do you know all sides? Use the cosine rule.

Do you need the area and have two sides and the included angle? Use the area rule.

 

Worked Example

Find the area of triangle PQR given:

$PQ = 6.5 cm$

$QR = 5.8 cm$

$RP = 4.1 cm$

We know all three sides but no angles.

 

 

Step 1: Find an Angle Using Cosine Rule

We choose to find angle $\angle PQR$ (opposite 4.1 cm):

$$\cos \angle PQR = \frac{6.5^2 + 5.8^2 - 4.1^2}{2(6.5)(5.8)}$$

$$\cos \angle PQR = \frac{42.25 + 33.64 - 16.81}{75.4} = \frac{59.08}{75.4} \approx 0.7837$$

$$\angle PQR = \cos^{-1}(0.7837) \approx 38.5\degree$$

 

Step 2: Use Area Formula

Now apply the area rule:

$$\text{Area} = \frac{1}{2}ab\sin C = \frac{1}{2}(6.5)(5.8)\sin(38.5^\circ)$$

$$= 18.85 \text{ cm}^2 (to \ 3 s.f.)$$

 

Summary

Choosing between sine and cosine rule comes down to knowing:

• Which parts of the triangle you already have
• What you’re trying to find