What is the Cosine Rule formula and how to use it with examples
The concept of cosine rule (also called the law of cosines) plays a key role in mathematics, especially in trigonometry and geometry. It is essential for finding missing sides or angles in any triangle—not just right-angled ones—and is a powerful tool for exam prep and real-life problem-solving.
What Is the Cosine Rule?
The cosine rule is a mathematical formula that relates the lengths of the sides of any triangle to the cosine of one of its angles. It allows you to calculate an unknown side or angle when you know other measurements. You’ll see this in action when solving for triangle sides in geometry, calculating distances in physics vectors, and during competitive exams like JEE or GCSE.
Key Formula for Cosine Rule
Here’s the standard formula for finding a side in a triangle:
\( c^2 = a^2 + b^2 - 2ab \cos(C) \)
Or, to rearrange for other sides/angles:
| Formula | Usage |
|---|---|
| \( a^2 = b^2 + c^2 - 2bc\cos(A) \) | Find side "a" if "b", "c" and angle A are known |
| \( b^2 = a^2 + c^2 - 2ac\cos(B) \) | Find side "b" with "a", "c", angle B |
| \( c^2 = a^2 + b^2 - 2ab\cos(C) \) | Find side "c" with "a", "b", angle C |
| \( \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \) | Find angle C when all sides are known |
When and How to Use the Cosine Rule
Use the cosine rule in the following cases:
- When you know two sides and the included angle (SAS case)
- When you know all three sides (SSS case) and need to find an angle
If the triangle does not have a right angle, or if the sine rule is not applicable, the cosine rule is your go-to formula. For more, read the Law of Sines to compare the methods.
Step-by-Step Illustration: Cosine Rule Example
Let's see how the cosine rule works with a step-by-step problem:
1. Write the formula:
\( c^2 = a^2 + b^2 - 2ab\cos(C) \ )
2. Plug in the given values:
\( c^2 = 8^2 + 11^2 - 2 \times 8 \times 11 \times \cos(37^\circ) \)
3. Calculate the values:
\( c^2 = 64 + 121 - 176 \times 0.7986 \) (cos 37° ≈ 0.7986)
4. Simplify:
\( c^2 = 185 - 140.53 = 44.47 \)
5. Take the square root:
\( c = \sqrt{44.47} \approx 6.67\, \text{cm} \)
Finding an Angle Using Cosine Rule
If all three sides of a triangle are known, you can find any angle:
1. Use the angle formula:
\( \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \)
2. Plug in values:
\( \cos(C) = \frac{5^2 + 7^2 - 9^2}{2 \times 5 \times 7} \)
\( = \frac{25 + 49 - 81}{70} \)
\( = \frac{-7}{70} = -0.1 \)
3. Find the angle using inverse cosine:
\( C = \arccos(-0.1) \approx 95.7^\circ \)
Calculator Tip & Quick Reference
You can use a scientific calculator for cosine rule problems. First calculate the cosine value, then multiply and subtract as per the formula. If you need to find an angle, use the “INV” or “cos-1” function for inverse cosine.
For instant calculation practice, try the online Trigonometry Table and formulas on Vedantu’s site!
Why Is the Cosine Rule Important?
For students, mastering the cosine rule helps to:
- Solve real exam questions for all board levels
- Calculate lengths or angles in non-right-angled triangles
- Apply trigonometry in higher-level maths, vectors, and physics problems
It’s also used in engineering, navigation, and even measuring distances indirectly. Learn more about triangle properties at Triangle and Its Properties.
Common Student Mistakes
- Mismatching the angle with the side in the formula
- Forgetting to take the correct inverse cosine when finding an angle
- Using the cosine rule when the sine rule applies (or vice versa)
- Errors in calculator input (degree/radian confusion)
Try These Yourself
- Find side c if a = 10, b = 6, and angle C = 40°
- Given triangle sides 7 cm, 8 cm, and 9 cm—find the largest angle
- When can you NOT use the cosine rule? Give an example
- Find angle A in triangle with a = 12, b = 9, c = 7
Want more problems? Explore Triangle Theorems for a variety of solved and practice questions.
Relation to Other Maths Concepts
The cosine rule links closely to other triangle topics. When the angle is 90°, it reduces to the Pythagorean Theorem. It's also fundamental when studying trigonometry, identities, and sine rule.
FAQ: Cosine Rule Essentials
- What is the cosine rule formula?
\( c^2 = a^2 + b^2 - 2ab\cos(C) \) - When to use the cosine rule?
When you know two sides and the angle between, or when all three sides are known. - Can I use cosine rule in right triangles?
Yes, but Pythagoras’ Theorem is simpler. When the angle is 90°, both give the same result. - How do I use it to find an angle?
Rearrange for cosine of the angle and use inverse cosine: \( \cos(C) = \frac{a^2+b^2-c^2}{2ab} \)
Classroom Tip
A helpful way to remember the cosine rule: it’s like Pythagoras, but you subtract “twice the product of the other sides times the cosine of the included angle.” In live Vedantu classes, teachers often draw and color-code the triangle to show which items go where in the formula for easy memory triggers.
Wrapping It All Up
We explored the cosine rule—from definition, formula, and examples, to connections with triangle properties and trigonometry. Practice questions and solved examples build real confidence for exams, and Vedantu’s platform gives free access to resources and doubts clearing in live sessions. Keep practicing, and you’ll master this important Maths tool!
Related Topics: Law of Sines | Trigonometric Identities | Triangle and Its Properties | Area of Triangle Formula
FAQs on Cosine Rule Explained for Any Triangle
1. What is the Cosine Rule in Maths?
The Cosine Rule (also called the Law of Cosines) states that in any triangle, the square of one side equals the sum of the squares of the other two sides minus twice their product multiplied by the cosine of the included angle. The formula is:
c² = a² + b² − 2ab cos C
It is used to find:
- An unknown side when two sides and the included angle are known (SAS).
- An unknown angle when all three sides are known (SSS).
2. What is the formula for the Cosine Rule?
The formula for the Cosine Rule is c² = a² + b² − 2ab cos C. In a triangle with sides a, b, c and opposite angles A, B, C:
- a² = b² + c² − 2bc cos A
- b² = a² + c² − 2ac cos B
- c² = a² + b² − 2ab cos C
3. How do you use the Cosine Rule to find a missing side?
To find a missing side using the Cosine Rule, substitute the known sides and included angle into c² = a² + b² − 2ab cos C and solve.
Example: If a = 5, b = 7, and C = 60°:
- c² = 5² + 7² − 2(5)(7)cos 60°
- c² = 25 + 49 − 70(0.5)
- c² = 74 − 35 = 39
- c = √39 ≈ 6.24
4. How do you use the Cosine Rule to find a missing angle?
To find a missing angle, rearrange the Cosine Rule to isolate cosine and then use the inverse cosine function. The formula becomes:
cos C = (a² + b² − c²) / 2ab
Example: If a = 6, b = 8, c = 10:
- cos C = (6² + 8² − 10²) / (2 × 6 × 8)
- cos C = (36 + 64 − 100) / 96
- cos C = 0 / 96 = 0
- C = cos⁻¹(0) = 90°
5. When should you use the Cosine Rule instead of the Sine Rule?
Use the Cosine Rule when you are given SAS (two sides and included angle) or SSS (three sides) information.
- Use Cosine Rule for SAS to find a side.
- Use Cosine Rule for SSS to find an angle.
- Use the Sine Rule for ASA, AAS, or SSA cases.
6. How is the Cosine Rule related to Pythagoras' theorem?
The Cosine Rule becomes Pythagoras' theorem when the included angle is 90°. If C = 90°, then cos 90° = 0, so the formula:
c² = a² + b² − 2ab cos C
becomes:
- c² = a² + b² − 0
- c² = a² + b²
7. Can the Cosine Rule be used for obtuse triangles?
Yes, the Cosine Rule works for acute, right, and obtuse triangles. When the angle is obtuse (greater than 90°), its cosine value is negative. In the formula c² = a² + b² − 2ab cos C, subtracting a negative number increases the result, giving a longer opposite side. This makes the Cosine Rule valid for all types of triangles.
8. What is an example problem using the Cosine Rule?
An example of the Cosine Rule is finding the third side when two sides and the included angle are known.
Example: a = 9, b = 12, C = 30°:
- c² = 9² + 12² − 2(9)(12)cos 30°
- c² = 81 + 144 − 216(0.866)
- c² ≈ 225 − 187.06 = 37.94
- c ≈ √37.94 ≈ 6.16
9. What are common mistakes when using the Cosine Rule?
Common mistakes when using the Cosine Rule include incorrect substitution and calculator errors.
- Using the wrong angle (it must be the included angle for SAS).
- Forgetting the − 2ab cos C term.
- Not squaring the correct sides.
- Using degree mode incorrectly (ensure calculator is in degrees if needed).
10. Why is the Cosine Rule important in trigonometry?
The Cosine Rule is important because it allows you to solve any triangle when enough information is given. It helps in:
- Finding unknown sides and angles.
- Solving non-right-angled triangles.
- Applications in geometry, physics, navigation, and engineering.
