Proof of tan(A + B) | Simple Explanation for Class 10

S6-SA2-0443

What is the Proof of tan (A + B) = (tan A + tan B) / (1 - tan A tan B) (introductory)?

Grade Level:

Class 10

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

Definition

What is it?

The proof of tan(A + B) = (tan A + tan B) / (1 - tan A tan B) shows how we can find the tangent of the sum of two angles (A and B) using the individual tangents of A and B. It's a fundamental identity in trigonometry, derived from the sum formulas for sine and cosine. This identity helps simplify complex trigonometric expressions and solve problems involving angles.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya, each telling you their 'angle of enthusiasm' for a new movie. Rohan's enthusiasm is tan A and Priya's is tan B. This formula helps you combine their individual enthusiasm levels to find the 'total enthusiasm' (tan(A+B)) for the movie, showing how their excitement blends together.

Worked Example

Step-by-Step

Let's prove tan(A + B) = (tan A + tan B) / (1 - tan A tan B) using sin(A+B) and cos(A+B) formulas.

Step 1: Recall the basic definition of tan x = sin x / cos x.
So, tan(A + B) = sin(A + B) / cos(A + B).

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Step 2: Recall the sum formulas for sine and cosine:
sin(A + B) = sin A cos B + cos A sin B
cos(A + B) = cos A cos B - sin A sin B

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Step 3: Substitute these into the tan(A + B) expression:
tan(A + B) = (sin A cos B + cos A sin B) / (cos A cos B - sin A sin B)

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Step 4: To get tan A and tan B, we need to divide everything by cos A cos B. This is a common trick!
Divide both the numerator and the denominator by cos A cos B:
tan(A + B) = [(sin A cos B + cos A sin B) / (cos A cos B)] / [(cos A cos B - sin A sin B) / (cos A cos B)]

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Step 5: Separate the terms in the numerator and denominator:
Numerator: (sin A cos B / cos A cos B) + (cos A sin B / cos A cos B)
Denominator: (cos A cos B / cos A cos B) - (sin A sin B / cos A cos B)

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Step 6: Simplify each term:
Numerator: (sin A / cos A) + (sin B / cos B) = tan A + tan B
Denominator: 1 - (sin A / cos A) * (sin B / cos B) = 1 - tan A tan B

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Step 7: Combine the simplified numerator and denominator:
tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

Answer: The identity is proven.

Why It Matters

Understanding this proof is crucial for solving problems in engineering, physics, and even AI/ML, where angles and rotations are common. Engineers use it to design structures and machines, while physicists apply it in wave mechanics and optics. In space technology, precise angle calculations are vital for satellite positioning and rocket trajectories.

Common Mistakes

MISTAKE: Assuming tan(A + B) = tan A + tan B. | CORRECTION: Remember that trigonometric functions don't distribute over addition like simple algebra. Always use the full identity: (tan A + tan B) / (1 - tan A tan B).

MISTAKE: Forgetting the sign in the denominator, writing (1 + tan A tan B) instead of (1 - tan A tan B). | CORRECTION: The minus sign in the denominator (1 - tan A tan B) is critical and comes directly from the cos(A+B) formula. Double-check your signs!

MISTAKE: Not dividing *all* terms by cos A cos B during the derivation, leading to incorrect simplification. | CORRECTION: When simplifying from sin and cos to tan, ensure you divide both terms in the numerator and both terms in the denominator by cos A cos B to correctly convert them.

Practice Questions

Try It Yourself

QUESTION: If tan A = 1/2 and tan B = 1/3, find the value of tan(A + B). | ANSWER: tan(A + B) = (1/2 + 1/3) / (1 - (1/2)*(1/3)) = (5/6) / (1 - 1/6) = (5/6) / (5/6) = 1

QUESTION: Given that A and B are acute angles, and tan A = 3/4, tan B = 4/3. Prove that A + B = 90 degrees. | ANSWER: tan(A + B) = (3/4 + 4/3) / (1 - (3/4)*(4/3)) = (9/12 + 16/12) / (1 - 1) = (25/12) / 0. Since the denominator is 0, tan(A+B) is undefined, which means A+B = 90 degrees (or pi/2 radians).

QUESTION: Derive the formula for tan(A - B) using the proof method for tan(A + B). (Hint: Replace B with -B). | ANSWER: tan(A - B) = (tan A - tan B) / (1 + tan A tan B). (Proof steps would involve using sin(A-B) and cos(A-B) or substituting -B for B in the tan(A+B) formula and using tan(-B) = -tan B).

MCQ

Quick Quiz

Which of the following is the correct identity for tan(A + B)?

(tan A - tan B) / (1 + tan A tan B)

(tan A + tan B) / (1 + tan A tan B)

(tan A + tan B) / (1 - tan A tan B)

(tan A - tan B) / (1 - tan A tan B)

The Correct Answer Is:

Option C is the correct identity for tan(A + B). Options A and D are for tan(A-B) or have incorrect signs. Option B has the wrong sign in the denominator.

Real World Connection

In the Real World

This concept is used in designing mobile tower networks. Engineers need to calculate the optimal angles for antenna placement to ensure maximum coverage and signal strength. Understanding how angles add up, using identities like tan(A+B), helps them predict signal interference and optimize network performance, ensuring your mobile data works smoothly even in busy areas.

Key Vocabulary

Key Terms

Identity: An equation that is true for all possible values of the variables | Numerator: The top part of a fraction | Denominator: The bottom part of a fraction | Trigonometry: The branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles | Derive: To obtain or deduce from a source or principle

What's Next

What to Learn Next

Next, you should explore the proof for tan(A - B) and then move on to understanding double angle and half angle formulas. These concepts build directly on the sum and difference identities, opening up more ways to solve complex trigonometric problems and prepare you for higher classes.