How to Use the Law of Cosines to Solve for Sides or Angles
The concept of Law of Cosines plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. From finding unknown triangle sides to tackling CBSE and JEE exam questions, this formula is essential for every learner of trigonometry.
What Is Law of Cosines?
The Law of Cosines, also known as the cosine rule, is a formula relating the lengths of the sides of any triangle to the cosine of one of its angles. You’ll find this concept applied in solving triangles, trigonometry class 11, and geometry problems where the Pythagorean theorem does not apply.
Key Formula for Law of Cosines
Here’s the standard formula:
For a triangle with sides a, b, c and opposite angles A, B, C:
c² = a² + b² − 2ab cos(C)
Similarly:
a² = b² + c² − 2bc cos(A)
b² = a² + c² − 2ac cos(B)
Cross-Disciplinary Usage
Law of Cosines is not only useful in Maths but also plays an important role in Physics (for force and vector problems), Computer Science (for graphics and gaming), and logical reasoning in everyday measurements. Students preparing for JEE or NEET will see its relevance in various types of triangle-based questions.
When to Use the Law of Cosines
The Law of Cosines is especially helpful in solving for unknown sides or angles in any triangle when:
- You know two sides and the included angle (SAS: Side-Angle-Side)
- You know all three sides (SSS: Side-Side-Side)
It is commonly used when the Law of Sines cannot be applied easily, especially when the angle is not opposite a known side.
Step-by-Step Illustration
- Suppose you have a triangle with a = 5 cm, b = 7 cm, and angle C = 60°.
Use Law of Cosines to find side c. - Apply the formula:
c² = 5² + 7² − 2 × 5 × 7 × cos(60°) - Calculate cos(60°):
cos(60°) = 0.5 - Substitute values:
c² = 25 + 49 − 2 × 5 × 7 × 0.5c² = 74 − 35c² = 39 - Find c:
c = √39 ≈ 6.24 cm
Speed Trick or Vedic Shortcut
Here’s a quick way to remember the Law of Cosines formula: Think of the Pythagoras theorem, but with an added "correction term" for non-right triangles.
Mnemonic: “Two squares minus double product times cosine!”
So, for c²: add a² + b², then subtract 2ab × cos(C).
This helps students quickly recall the formula even under exam pressure.
Try These Yourself
- A triangle has sides 4 cm and 6 cm with an included angle of 45°. Find the third side using Law of Cosines.
- Given triangle sides a = 5 cm, b = 8 cm, c = 7 cm, find angle C.
- Can you apply Law of Cosines to an equilateral triangle? What do you notice?
Frequent Errors and Misunderstandings
- Forgetting to use the correct included angle in the formula.
- Mixing up Law of Sines and Law of Cosines situations.
- Making calculator mistakes with cosine values in degrees vs radians.
Relation to Other Concepts
The idea of Law of Cosines connects closely with trigonometric formulas, triangle properties, and extends the Pythagorean theorem to all triangles (not just right-angled ones). Mastering this helps with understanding the Properties of Triangles and more advanced geometry and vector topics.
Law of Cosines vs Law of Sines
| Law of Cosines | Law of Sines |
|---|---|
| Use for SSS or SAS cases; relates sides and included angle using cosine. | Use for ASA or AAS cases; relates ratios of sides and their opposite angles using sine. |
| a² = b² + c² – 2bc cos(A) | a/sin A = b/sin B = c/sin C |
| Solves for unknown sides or angles when direct ratios are not available. | Best when an angle and its opposite side are both known. |
Applications in Real Life
Law of Cosines is important not just in exams but also in real-world applications like:
- Surveying land when you cannot measure directly across rough terrain
- Navigation and GPS calculations for ships and aircraft
- Physics problems involving vector resolution and forces
- Designing objects or structures where precise triangle measurements are needed
You'll often see questions about these cases in JEE, CBSE, and Olympiad exams as well.
Classroom Tip
A quick way to remember Law of Cosines is: “For any side squared, add the other two sides squared, subtract double their product times the cosine of the included angle.” Vedantu’s teachers often use diagrams and real-life sketches to show why this works and help students visualize triangle relationships.
Quick Reference Table
| To Find | Law of Cosines Variant | Formula |
|---|---|---|
| Unknown side (c) | SAS Case | c² = a² + b² − 2ab cos(C) |
| Unknown angle (C) | SSS Case | cos(C) = (a² + b² − c²) / 2ab |
We explored Law of Cosines—from definition, formula, step-by-step example, errors, and advanced applications. Continue practicing with Vedantu to become confident in solving triangles using this concept.
Related topics to boost your understanding:
FAQs on Law of Cosines: Formula, Proof, and Problem-Solving for Triangles
1. What is the Law of Cosines and when is it used in Class 11 Maths?
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. As per the CBSE syllabus, it is essential for solving triangles in two specific scenarios:
- When you know the lengths of all three sides of a triangle (SSS case).
- When you know the lengths of two sides and the angle between them (SAS case).
2. What are the standard formulas for the Law of Cosines?
The Law of Cosines can be written in three forms, depending on which side or angle you need to find. For a triangle with sides a, b, c and opposite angles A, B, C respectively, the formulas are:
- a² = b² + c² - 2bc cos(A)
- b² = a² + c² - 2ac cos(B)
- c² = a² + b² - 2ab cos(C)
3. How do you apply the Law of Cosines to find an unknown side of a triangle?
To find an unknown side, you must know the lengths of the other two sides and the angle included between them (SAS). For example, to find side c, you would use the formula c² = a² + b² - 2ab cos(C). Simply substitute the known values for sides a and b, and the angle C, then calculate the result and take its square root to find the length of c.
4. How can the Law of Cosines be used to find an unknown angle in a triangle?
To find an unknown angle, you must know the lengths of all three sides (SSS). You need to rearrange the formula. For instance, to find angle C, you would rearrange the formula c² = a² + b² - 2ab cos(C) to solve for cos(C):
cos(C) = (a² + b² - c²) / 2ab
After calculating the value of cos(C), you use the inverse cosine function (cos⁻¹) on your calculator to find the measure of angle C.
5. What is the key difference between using the Law of Sines and the Law of Cosines?
The key difference lies in the information you have about the triangle. You should use:
- The Law of Sines when you know a side and its opposite angle, plus one other piece of information (like another side or angle). It relies on having an angle-side pair.
- The Law of Cosines when you do not have an angle opposite a known side. It is specifically for cases where you know two sides and the included angle (SAS) or all three sides (SSS).
6. How is the Pythagorean theorem a special case of the Law of Cosines?
The Pythagorean theorem (a² + b² = c²) only applies to right-angled triangles. The Law of Cosines is a more general formula that works for any triangle. If you apply the Law of Cosines to a right-angled triangle where angle C is 90°, the term cos(C) becomes cos(90°), which is 0. The formula c² = a² + b² - 2ab cos(C) then simplifies to c² = a² + b² - 0, which is exactly the Pythagorean theorem.
7. Does the Law of Cosines work for all types of triangles, including obtuse ones?
Yes, the Law of Cosines works perfectly for all types of triangles: acute, right-angled, and obtuse. The cosine function handles the different angle types automatically. For an obtuse angle (between 90° and 180°), the cosine value is negative. This correctly adjusts the formula by making the term '-2ab cos(C)' positive, resulting in a longer third side, which is expected in an obtuse triangle.
8. What is the most common mistake students make when applying the Law of Cosines?
The most frequent error is a mismatch between the side being calculated and the angle used in the cosine function. For the formula to be correct, the side on the left side of the equation must be the one opposite the angle used in the cosine term on the right. For example, if you are calculating c², you must use the cosine of angle C, not A or B.
9. Why is the Law of Cosines important in Physics?
In Physics, the Law of Cosines is crucial for problems involving vectors. When two vectors (like forces or velocities) are added, they form a triangle with their resultant vector. The Law of Cosines is used to calculate the magnitude of the resultant vector when the magnitudes of the two initial vectors and the angle between them are known. It is a direct mathematical tool for solving problems in mechanics and kinematics.
10. What are some real-world examples where the Law of Cosines is applied?
The Law of Cosines is used in many practical fields beyond the classroom. Some important examples include:
- Surveying: To calculate the distance between two points separated by an obstacle like a river or a hill.
- Navigation and Aviation: To determine the distance and bearing for a plane or ship to travel between two points.
- Engineering and Architecture: For designing structures, such as the triangular trusses in bridges and roofs, ensuring stability and correct dimensions.
- Astronomy: To find the distances between celestial bodies by forming a triangle between Earth and two other objects.
11. What is the fundamental concept behind the proof of the Law of Cosines?
The most common proof of the Law of Cosines uses the distance formula from coordinate geometry. The triangle is placed on a Cartesian plane with one vertex at the origin (0,0) and another on the x-axis. The coordinates of the third vertex are then expressed using trigonometric functions (e.g., (b cos C, b sin C)). By applying the distance formula between the two vertices not at the origin and simplifying the resulting algebraic expression, you directly derive the Law of Cosines formula.
