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?
The Law of Cosines, also known as the cosine rule, is a fundamental trigonometric formula that relates the lengths of the sides of any triangle to the cosine of one of its angles. It's particularly useful for solving triangles when you know two sides and the included angle (SAS) or all three sides (SSS).
2. What is the formula for the Law of Cosines?
The formula is expressed in several ways, depending on which side and angle you're solving for. The most common form is: c² = a² + b² - 2ab cos(C), where c is the length of the side opposite angle C, and a and b are the lengths of the other two sides.
3. How do I use the Law of Cosines to find the length of a side?
If you know two sides (a and b) and the included angle (C), simply substitute the values into the formula c² = a² + b² - 2ab cos(C). Solve for c by taking the square root of both sides. Remember to consider the units of measurement.
4. How do I use the Law of Cosines to find an angle?
If you know all three sides (a, b, and c), you can rearrange the formula to solve for the cosine of the angle. For example, to find angle C, use: cos(C) = (a² + b² - c²) / 2ab. Then, use the inverse cosine function (cos⁻¹) on your calculator to find the angle measure.
5. When should I use the Law of Cosines instead of the Law of Sines?
Use the Law of Cosines when you have either: 1) two sides and the included angle (SAS), or 2) all three sides (SSS). Use the Law of Sines when you have two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA – be mindful of the ambiguous case).
6. Can the Law of Cosines be used for right-angled triangles?
Yes! In a right-angled triangle, where one angle is 90°, the Law of Cosines simplifies to the Pythagorean theorem (a² + b² = c²). The cosine of 90° is 0, making the term -2ab cos(C) equal to 0.
7. What are some common mistakes when using the Law of Cosines?
Common mistakes include: incorrect substitution of values into the formula; using the wrong units; forgetting to take the square root when solving for a side; and misinterpreting calculator results (especially when dealing with inverse cosine).
8. How is the Law of Cosines derived?
The derivation involves using the distance formula in coordinate geometry and applying trigonometric identities. Several geometrical methods can also be used to demonstrate the Law of Cosines. Consult your textbook or online resources for a detailed proof.
9. Does the Law of Cosines work for obtuse triangles?
Yes, the Law of Cosines works for all types of triangles – acute, right-angled, and obtuse. The cosine function handles the different angle ranges appropriately.
10. What are some real-world applications of the Law of Cosines?
The Law of Cosines has many applications in fields like surveying, navigation, and engineering. It's used to calculate distances, angles, and positions in various scenarios where triangles are involved.
11. How does the Law of Cosines relate to the Pythagorean Theorem?
The Pythagorean Theorem is a special case of the Law of Cosines that applies only to right-angled triangles. When the angle C is 90 degrees, cos(C) = 0, and the Law of Cosines reduces to a² + b² = c².
12. What are some tips for solving problems using the Law of Cosines?
• Draw a clear diagram of the triangle, labeling all known and unknown values.
• Carefully substitute the known values into the correct formula.
• Show all your working steps clearly.
• Check your answer using a different method if possible.
• Always double-check your calculations for errors.
