Cosine Rule

The cosine rule or the law of cosines has been a game-changer in the field of trigonometry. When the triangles are not right-angled, so the Pythagorean theorem doesn’t work, or when the sine rule just doesn’t cut it, this powerful rule steps in to save the day. By connecting the sides of the triangle to the cosine of one of its angles, you have a shortcut to finding the remaining sides and angles. 

You can think of the cosine rule as a more flexible version of the Pythagorean theorem, as it works for any triangle. The cosine rule formula can help with geometry problems as well as physics vectors. It bridges the gap between theoretical knowledge and real-world applications. Let’s learn more about it: 

1.0What Is the Cosine Rule?

The cosine rule is one of the wonders of trigonometry, connecting the sides of the triangle to the cosine of one of its angles. But what makes it so powerful? The formula works even when the triangle doesn’t have a right angle. Sharp angle? Wide angle? Doesn’t matter. If your triangle’s not right, use this!

Here’s the cosine rule formula for finding a missing side:

c² = a² + b² - 2ab.cos(C)

Where:

• a and b are the sides of the triangle
• C is the angle on the opposite of side c

To find an angle, you  can rearrange it like this:

Cos(C) = (a² + b² - c²)/2ab

2.0When to Use the Cosine Rule?

Not sure if the cosine rule is the right move? Here’s when to use cosine rule:

• Got two sides and the angle between them? That’s the SAS case, and cosine rule is perfect for this.
• Know all three sides and need to find an angle? That’s the SSS case, and cosine rule would work.
• If it’s a right-angled triangle, stick with good old Pythagoras instead.

3.0Step-by-Step Example: Finding a Side

Step-by-Step Example: Finding a Side with the Cosine Rule

Problem:
In triangle ABC, side b = 9 cm, side c = 12 cm, and angle A = 65°. Find the length of side a.

Step 1: Write the Cosine Rule Formula

To find a side when two other sides and the included angle are known, use:

a² = b² + c² - 2bc.cos(A)

Step 2: Plug in the Known Values

a² = 9² + 12² - 2(9)(12).cos(65°)

Step 3: Simplify the Equation

a² = 81 + 144 - 216*0.4226 (Using a calculator for cos(65°))

a² = 225 - 91.286

a² = 133.7184

Step 4: Take the Square Root

a = 11.56 cm

Final Answer:

The missing side a ≈ 11.56 cm

4.0Finding an Angle Using Cosine Rule

Let’s try a different scenario—this time, you know all three sides.

Problem:

In triangle XYZ, the sides are:

• XY = 8 cm
• YZ = 11 cm
• ZX = 13 cm

Find the angle ∠Y (the angle opposite side ZX).

Solution:

We’ll use the Cosine Rule Formula for Angles:

cos⁡(A) = (b² + c² − a²)/2bc​

Step 1: Plug in the known values

cos⁡(Y) = (8²+11²−13²)/2×8×11

Step 2: Simplify the equation

cos⁡(Y) = (64+121−169)/2×8×11

 cos⁡(Y) = (185−169)/176

cos⁡(Y) = 16/176

cos⁡(Y) = 0.0909

Step 3: Take the inverse cosine

Use a calculator to find the angle:

Y = cos⁡⁻¹(0.0909)

Y ≈ 84.78

Final Answer:

The angle ∠Y ≈ 84.8°

5.0Calculator Tips

Doing a cosine rule question?

• Switch that calculator to DEG mode
• cos = find a side
• cos⁻¹ = find an angle
• Don’t mess up your parentheses.
• If you need help, you can use an interactive trig calculator online!

6.0Why the Cosine Rule Matters

Mastering the law of cosines prepares you for more than exams:

• In Physics, you’ll work with vectors and forces.
• In Engineering, you can calculate distances with confidence.
• In surveying & navigation, it makes Indirect measurement easy.

This formula is part of your foundation for STEM success.

Common Student Mistakes

• Don’t Let These Mistakes Ruin Your Cosine Rule Game
• Wrong triangle labelling = disaster
• No inverse cosine = no angle
• Using the cosine rule when sine would've been a breeze
• Wrong calculator mode = wrong life choices

Master Tip: Label before you calculate. Every. Single. Time.

7.0Try These Yourself

Want to test your skills? Try solving these cosine rule examples on your own:

1. Find side c if a = 10, b = 6, and angle C = 40^∘
2. Given triangle sides 7 cm, 8 cm, and 9 cm, find the largest angle.
3. In triangle ABC, a = 12, b = 9, c = 7. Find angle A.