Geometry

Cosine rule

Open the angle. Watch the third side.

Predict, test and explain

Step 1 of 3

Predict

Set sides a = 3 and b = 4 with included angle C = 90°. What is side c?

c2=a2+b2−2abcos⁡Cc^2=a^2+b^2-2ab\cos C
A≈43.9∘A \approx 43.9^\circ
B≈76.1∘B \approx 76.1^\circ
C=60∘C = 60^\circ
a=5a = 5
b=7b = 7
c≈6.24c \approx 6.24

Scroll the diagram sideways to see every label.

a2+b2=74a^2+b^2 = 74−2abcos⁡C=−35-2ab\cos C = -35c≈6.24c \approx 6.24

InvestigateAt which angle does this become Pythagoras?

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Two sides of a triangle are 5 cm and 7 cm. Their included angle is 60 degrees. Find the third side exactly.

Hint

Use the cosine rule with the angle between the two known sides.

Reveal answer

The third side is the positive square root of 39, in cm.

c2=52+72−2(5)(7)cos⁡60∘=39,c=39 cmc^2=5^2+7^2-2(5)(7)\cos60^\circ=39,\qquad c=\sqrt{39}\,\text{cm}

Build an explanation

Hint 1

Find the angle between the two known sides.

Hint 2

The third side changes as that included angle opens.

Hint 3

Use c² = a² + b² − 2ab cos C when C is between a and b.

Worked example

This example uses fixed values, separate from the diagram controls.

Two sides are 5 cm and 7 cm, with included angle 60°. Find the third side.

  1. c² = 5² + 7² − 2 × 5 × 7 × cos 60°.
  2. c² = 25 + 49 − 35 = 39.
  3. Take the positive square root.

Answer: √39 cm, approximately 6.24 cm.

Connect this idea

Watch out: The cosine-rule angle must be the included angle between the two sides in the product.