When to Use the Cosine Rule vs Sine Rule in Triangle Problems
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: Formula, Proof, and How to Use It in Triangles
1. What is the Cosine Rule and what is its main purpose in Maths?
The Cosine Rule, also known as the Law of Cosines, is a fundamental formula in trigonometry that establishes a relationship between the lengths of the sides of any triangle and the cosine of one of its angles. Its main purpose is to find a missing side or angle in a non-right-angled triangle when other specific measurements are known.
2. What are the two main formulas for the Cosine Rule?
The Cosine Rule can be written in two ways, depending on whether you need to find a side or an angle:
- To find a missing side (e.g., side 'c'): The formula is c² = a² + b² - 2ab cos(C). Here, 'a' and 'b' are the other two sides, and 'C' is the angle included between them.
- To find a missing angle (e.g., angle 'C'): The formula is rearranged to cos(C) = (a² + b² - c²) / 2ab.
3. When should I use the Cosine Rule instead of the Sine Rule in a triangle problem?
You should use the Cosine Rule in two specific scenarios:
- When you know the lengths of two sides and the included angle (SAS).
- When you know the lengths of all three sides and need to find an angle (SSS).
In contrast, the Sine Rule is used when you know a pair of an angle and its opposite side, such as having two angles and one side (AAS/ASA).
4. How is the Cosine Rule proven using basic geometry?
The standard proof of the Cosine Rule involves dividing a triangle into two right-angled triangles by drawing a perpendicular from one vertex to the opposite side. By applying the Pythagorean Theorem to both of these new right-angled triangles and using the basic definition of cosine (cos = adjacent/hypotenuse), the resulting algebraic expressions can be combined and simplified to derive the formula c² = a² + b² - 2ab cos(C).
5. What is the relationship between the Cosine Rule and the Pythagorean Theorem?
The Pythagorean Theorem is actually a special case of the Cosine Rule. If you apply the Cosine Rule to a right-angled triangle where the angle C is 90°, the value of cos(90°) is 0. This makes the '- 2ab cos(C)' part of the formula disappear, simplifying c² = a² + b² - 2ab(0) to just c² = a² + b². This demonstrates that the Cosine Rule is a more general formula that works for all triangles.
6. What does it mean if I get a negative result when calculating cos(C) with the Cosine Rule?
A negative result for cos(C) does not mean your calculation is incorrect. It is an important indicator that the angle 'C' you are solving for is an obtuse angle (an angle greater than 90° but less than 180°). The cosine function is negative for angles in this range. Simply use the inverse cosine function (cos⁻¹) on your calculator to find the correct obtuse angle value.
7. What are some real-world examples where the Cosine Rule is applied?
The Cosine Rule is essential in many practical fields. Some key applications include:
- Navigation and Surveying: To calculate the distance between two points or a ship's bearing when direct measurement is not possible.
- Engineering: To determine forces and angles in truss structures and mechanical systems.
- Physics: To find the magnitude of a resultant vector when adding two vectors (e.g., forces or velocities) that act at an angle to each other.
8. Beyond just finding sides and angles, in what other types of geometry problems is the Cosine Rule useful?
The Cosine Rule is a versatile tool for solving more complex geometric problems where a side or angle is not the final answer. For example, it is a crucial step for:
- Finding the length of a median in a triangle.
- Solving problems involving quadrilaterals by dividing them into two triangles to find a diagonal's length.
- Determining the nature of a triangle's angles (acute or obtuse) without calculating them directly.
9. Can the Cosine Rule be proven using vectors, and what is the advantage of this method?
Yes, an elegant and concise proof of the Cosine Rule uses the vector dot product. By representing the sides of a triangle as vectors and using the property that the dot product **a** · **b** = |a||b|cos(θ), the formula can be derived quickly. The advantage of the vector proof is its efficiency and its power in connecting geometric properties to algebraic vector operations, a concept frequently used in higher-level Maths and Physics as per the CBSE curriculum for 2025-26.
