What is the Law of Cosines? | Simple Explanation for Class 10

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What is the Law of Cosines (introductory)?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The Law of Cosines is a rule that connects the lengths of the sides of ANY triangle (not just right-angled ones) to the cosine of one of its angles. It's like an extended version of the Pythagoras theorem that works for all triangles, helping us find unknown side lengths or angles.

Simple Example

Quick Example

Imagine you are flying a kite. You know the length of your kite string (one side) and the distance your friend is standing from you (another side). If you also know the angle between your string and the ground, the Law of Cosines can help you find the actual height of the kite from your friend's position without needing to climb up!

Worked Example

Step-by-Step

Let's say we have a triangle ABC. Side 'a' is opposite angle A, side 'b' opposite angle B, and side 'c' opposite angle C. Suppose side b = 8 cm, side c = 5 cm, and the angle A between them is 60 degrees. Let's find the length of side 'a'.

1. **Write the formula:** The Law of Cosines for finding side 'a' is: a^2 = b^2 + c^2 - 2bc * cos(A)
---2. **Substitute the values:** a^2 = (8)^2 + (5)^2 - 2 * 8 * 5 * cos(60 degrees)
---3. **Calculate squares:** a^2 = 64 + 25 - 2 * 8 * 5 * cos(60 degrees)
---4. **Know cos(60 degrees):** We know that cos(60 degrees) = 0.5
---5. **Continue substituting and multiplying:** a^2 = 64 + 25 - 2 * 8 * 5 * 0.5
---6. **Simplify:** a^2 = 89 - 40
---7. **Calculate a^2:** a^2 = 49
---8. **Find 'a':** a = sqrt(49) = 7 cm

**Answer:** The length of side 'a' is 7 cm.

Why It Matters

The Law of Cosines is crucial in fields like AI/ML for computer vision, helping robots understand distances and shapes. In engineering, it's used to design stable structures like bridges and buildings. Space Technology uses it to calculate distances between celestial bodies, guiding rockets and satellites precisely.

Common Mistakes

MISTAKE: Forgetting the minus sign and the '2bc' part, making it look like Pythagoras theorem (a^2 = b^2 + c^2 + cos(A)) | CORRECTION: Remember the full formula: a^2 = b^2 + c^2 - 2bc * cos(A). The '- 2bc * cos(A)' term is essential for non-right triangles.

MISTAKE: Using the wrong angle for the given sides, e.g., using angle B when calculating side 'a' from sides 'b' and 'c'. | CORRECTION: When finding side 'a', you must use the angle 'A' (the angle opposite side 'a'). Similarly, for side 'b', use angle 'B'.

MISTAKE: Incorrectly calculating the cosine value, especially for angles like 90 degrees or 180 degrees. | CORRECTION: Always double-check the cosine values for common angles (like cos(0)=1, cos(60)=0.5, cos(90)=0, cos(180)=-1) or use a calculator correctly.

Practice Questions

Try It Yourself

QUESTION: In triangle PQR, side p = 10 cm, side q = 12 cm, and the angle R between them is 30 degrees. Find the length of side r. | ANSWER: r^2 = p^2 + q^2 - 2pq * cos(R) = 10^2 + 12^2 - 2 * 10 * 12 * cos(30) = 100 + 144 - 240 * (sqrt(3)/2) = 244 - 120 * sqrt(3). So, r = sqrt(244 - 120 * sqrt(3)) approximately 6.01 cm.

QUESTION: A triangular garden has sides of 7 meters, 9 meters, and 13 meters. Find the largest angle of the garden. (Hint: The largest angle is opposite the longest side.) | ANSWER: Let sides be a=7, b=9, c=13. We need to find angle C. c^2 = a^2 + b^2 - 2ab * cos(C). 13^2 = 7^2 + 9^2 - 2 * 7 * 9 * cos(C). 169 = 49 + 81 - 126 * cos(C). 169 = 130 - 126 * cos(C). 39 = -126 * cos(C). cos(C) = -39/126 approximately -0.3095. C = arccos(-0.3095) approximately 108.03 degrees.

QUESTION: Two auto-rickshaws start from the same stand. One travels 15 km North, and the other travels 20 km on a path 45 degrees East of North. How far apart are the two auto-rickshaws? | ANSWER: Let the stand be A. First auto travels to B (15 km North), second to C (20 km, 45 degrees East of North). We need to find BC. Angle BAC = 45 degrees. Using Law of Cosines: BC^2 = AB^2 + AC^2 - 2 * AB * AC * cos(A). BC^2 = 15^2 + 20^2 - 2 * 15 * 20 * cos(45). BC^2 = 225 + 400 - 600 * (1/sqrt(2)). BC^2 = 625 - 600/sqrt(2) approximately 625 - 424.26 = 200.74. BC = sqrt(200.74) approximately 14.17 km.

MCQ

Quick Quiz

Which of the following formulas correctly represents the Law of Cosines to find side 'b' in triangle ABC?

b^2 = a^2 + c^2 + 2ac * cos(B)

b^2 = a^2 + c^2 - 2ac * cos(B)

b^2 = a^2 + c^2 - 2ac * sin(B)

b = a + c - 2ac * cos(B)

The Correct Answer Is:

Option B is correct. The Law of Cosines involves the square of the sides, a minus sign, and the cosine of the angle opposite the side being calculated. Options A and C have incorrect signs or trigonometric functions. Option D does not have the sides squared.

Real World Connection

In the Real World

From helping ISRO scientists calculate the trajectory of rockets to ensuring your mobile phone's GPS accurately pinpoints your location in a crowded Indian market, the Law of Cosines plays a silent but vital role. It's also used by architects to design buildings and by land surveyors when measuring plots of land for new construction projects.

Key Vocabulary

Key Terms

TRIANGLE: A polygon with three sides and three angles. | COSINE: A trigonometric ratio relating an angle of a right-angled triangle to the ratio of the adjacent side to the hypotenuse. In the Law of Cosines, it's extended to any triangle. | SIDES: The line segments forming the boundary of a triangle. | ANGLES: The space between two intersecting lines or surfaces at or close to the point where they meet. | PYTHAGORAS THEOREM: A fundamental relation in Euclidean geometry among the three sides of a right triangle.

What's Next

What to Learn Next

Once you're comfortable with the Law of Cosines, you should explore the 'Law of Sines'. It's another powerful tool for solving triangles and often works hand-in-hand with the Law of Cosines, especially when you have different sets of known information. Keep practicing!