Cosine Rule
The cosine rule finds a missing side or angle. If you know two sides and the angle between them, or all three sides, use a² = b² + c² − 2bc cos A. Use it in non-right-angled triangles; right-angled ones use Pythagoras.
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The Cosine Rule
- The cosine rule is used to find a missing side or angle in any triangle.
- It is used when the triangle is not right-angled (otherwise use Pythagoras’ theorem).
Example: The Cosine Rule for Finding a Side
- If you know two sides and the angle between them, you can find the third side.
- The rule is , where a is the unknown side and A is the angle between b and c.
Rearranging the Cosine Rule to Find an Angle
- If you know all three sides, you can find a missing angle.
- Rearrange to . Once you find , you can find angle A.
Example: Finding an Angle Using the Cosine Rule
- Substitute the three side lengths into the rearranged formula.
- Once you find , you can find angle F.
- Use on your calculator to find the angle in degrees.
Practice Questions
Test your understanding
Which of the following is the correct Cosine Rule formula?
Correct! 🎉 +10 pointsNot quite right
The correct formula for the Cosine Rule is , where and are the sides of the triangle, and is the angle between them.
What does the Cosine Rule help you find in a triangle?
Correct! 🎉 +10 pointsNot quite right
The Cosine Rule is used to find missing sides in a triangle when we know two sides and the included angle. It can also be used to find missing angles when we know all three sides of the triangle.
In a triangle, sides a , b , and the angle between them is . What is the third side?
Correct! 🎉 +20 pointsNot quite right
Using the Cosine Rule, we calculate . Since , we have . Therefore, .
In a triangle, sides a , b , and the angle between them is . What is the third side?
Correct! 🎉 +20 pointsNot quite right
Using the Cosine Rule formula , where , , and , we calculate . Since , we have . Therefore, .
In a triangle, sides a , b , and the angle between them is . What is the third side?
Correct! 🎉 +20 pointsNot quite right
Using the Cosine Rule formula , where , , and , we calculate . Since , we have . Therefore, .
In a triangle, sides a , b , and c . Find the angle between sides a and b.
Correct! 🎉 +30 pointsNot quite right
Using the Cosine Rule: , where , , and . Substituting the values, we get , which simplifies to . We find , and using the inverse cosine function, we get .
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Interactive Activity
Practice using the Cosine Rule to find a missing side or angle of a non-right-angled triangle
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Students Also Ask
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Use the cosine rule when you know two sides of a triangle and the angle between them, so you can find the third side. Also use it when you know all three sides and want to find one of the angles. It works in any triangle, not just right-angled ones.
The cosine rule is an extension of Pythagoras' theorem. When the angle between the two known sides is 90°, cos(90°) equals 0, so the term 2bc × cos(A) becomes zero and the formula collapses to a² = b² + c². This is exactly Pythagoras' theorem for a right-angled triangle.
The cosine rule has two equivalent forms. The standard form, a² = b² + c² − 2bc × cos(A), finds a missing side when you know two sides and the included angle. The rearranged form, cos(A) = (b² + c² − a²) ÷ (2bc), finds a missing angle when you know all three sides.
No, you do not need to label the triangle when finding an angle. Simply subtract the square of the side opposite the angle you are finding the cosine of, then divide by twice the product of the other two sides. The labelling takes care of itself.
Side lengths in a triangle must be positive, so the negative root is not a valid length. When solving x² = 19 for example, the square root gives both +√19 and −√19, but only the positive value of about 4.4 makes sense as a side length in a real triangle.
Yes, the cosine rule works for every triangle, including right-angled ones. In a right-angled triangle with the 90° angle as A, cos(90°) equals 0 and the formula reduces to Pythagoras' theorem. For non-right-angled triangles, the cosine of the angle adds the extra correction term.