What is the Proof of cos(A+B)? | Simple Explanation for Class 10

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What is the Proof of cos (A + B) = cos A cos B - sin A sin B (introductory)?

Grade Level:

Class 10

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

Definition

What is it?

The proof of cos (A + B) = cos A cos B - sin A sin B shows us how to derive this important trigonometric identity using geometric methods, usually involving a unit circle and coordinate geometry. It explains why the cosine of the sum of two angles equals the product of their individual cosines minus the product of their individual sines.

Simple Example

Quick Example

Imagine you are calculating the distance an airplane travels. If the plane changes direction twice, the total change in its horizontal position can be found by adding the angles of its turns. This formula helps break down that combined turn into simpler calculations, just like how combining two discounts on your favourite online shopping app changes the final price.

Worked Example

Step-by-Step

Let's prove cos(A+B) = cos A cos B - sin A sin B using a geometric approach (conceptual steps, not full derivation):
1. Draw a unit circle (radius 1) with its center at the origin (0,0) on a coordinate plane.
2. Mark a point P on the circle such that the angle it makes with the positive x-axis is A. Its coordinates will be (cos A, sin A).
3. Mark another point Q on the circle such that the angle it makes with the positive x-axis is (A+B). Its coordinates will be (cos(A+B), sin(A+B)).
4. Now, rotate the entire setup so that the point P lies on the positive x-axis. This means we are rotating by an angle -A.
5. After this rotation, the new coordinates of P will be (1, 0) and the new coordinates of Q will be (cos B, -sin B) (because the angle from the x-axis to Q is now B, but measured clockwise, making the sine negative).
6. The distance between the original point (1,0) and Q (cos(A+B), sin(A+B)) is the same as the distance between the rotated point (1,0) and rotated Q (cos B, -sin B). Using the distance formula, we can equate these squared distances.
7. (cos(A+B) - 1)^2 + (sin(A+B) - 0)^2 = (cos B - 1)^2 + (-sin B - 0)^2
8. Expanding and simplifying both sides using sin^2(x) + cos^2(x) = 1, we eventually arrive at cos(A+B) = cos A cos B - sin A sin B.

Why It Matters

This formula is fundamental in fields like Physics for analyzing wave motion and signal processing, and in Engineering for designing structures and electronic circuits. Engineers use it to understand how forces combine or how different signals interact, making it crucial for careers in space technology or even designing your next smartphone.

Common Mistakes

MISTAKE: Assuming cos(A+B) is simply cos A + cos B. | CORRECTION: Trigonometric functions don't distribute over addition like regular numbers. Always use the full identity: cos(A+B) = cos A cos B - sin A sin B.

MISTAKE: Forgetting the minus sign in the formula, writing cos(A+B) = cos A cos B + sin A sin B. | CORRECTION: Remember, for cos(A+B), the sign between the two terms is minus. Think of it as 'C for Cosine, C for Change (the sign)'.

MISTAKE: Confusing the formula for cos(A+B) with sin(A+B). | CORRECTION: The cos(A+B) formula involves both cos and sin terms, with a minus sign. The sin(A+B) formula involves alternating sin and cos terms with a plus sign (sin A cos B + cos A sin B).

Practice Questions

Try It Yourself

QUESTION: If cos A = 3/5 and sin B = 5/13, and A and B are acute angles, find cos(A+B). (Hint: Find sin A and cos B first.) | ANSWER: cos(A+B) = 16/65

QUESTION: Given cos 75 degrees = cos (45 degrees + 30 degrees), use the identity to find its value. (You know sin/cos of 30 and 45 degrees). | ANSWER: cos 75 degrees = (sqrt(6) - sqrt(2))/4

QUESTION: If A = 60 degrees and B = 30 degrees, verify that cos(A+B) = cos A cos B - sin A sin B. | ANSWER: LHS = cos(60+30) = cos 90 = 0. RHS = cos 60 cos 30 - sin 60 sin 30 = (1/2)(sqrt(3)/2) - (sqrt(3)/2)(1/2) = sqrt(3)/4 - sqrt(3)/4 = 0. Since LHS = RHS, the identity is verified.

MCQ

Quick Quiz

Which of the following is the correct identity for cos(X+Y)?

cos X cos Y + sin X sin Y

cos X cos Y - sin X sin Y

sin X cos Y + cos X sin Y

sin X cos Y - cos X sin Y

The Correct Answer Is:

The correct identity for cos(X+Y) is cos X cos Y - sin X sin Y. Option A is for cos(X-Y), and options C and D are for sin(X+Y) and sin(X-Y) respectively.

Real World Connection

In the Real World

In satellite communication, ISRO scientists use these trigonometric identities to precisely calculate the position and trajectory of satellites. When a satellite's antenna needs to be pointed at a ground station, the angles involved are often sums or differences of other angles, and this formula helps ensure accurate signal transmission, just like how your GPS app uses complex math to show your exact location.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all possible values of its variables. | UNIT CIRCLE: A circle with a radius of 1 unit, centered at the origin of a coordinate system. | COORDINATE GEOMETRY: The study of geometry using coordinates to represent points and shapes. | TRIGONOMETRY: The branch of mathematics dealing with the relations between the sides and angles of triangles. | ACUTE ANGLE: An angle measuring less than 90 degrees.

What's Next

What to Learn Next

Next, explore the proof of sin(A+B) = sin A cos B + cos A sin B. It uses a similar geometric approach and builds directly on your understanding of angle addition formulas. Keep going, you're building a strong foundation for advanced math!