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

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

Grade Level:

Class 10

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

Definition

What is it?

The proof of sin (A + B) = sin A cos B + cos A sin B shows how we can find the sine of the sum of two angles (A and B) using the sines and cosines of the individual angles. This formula is derived using geometric principles, specifically by drawing angles and their projections on a coordinate plane or within a right-angled triangle setup.

Simple Example

Quick Example

Imagine you have two cricket shots, one at 30 degrees and another at 45 degrees to the pitch. If you wanted to combine their 'forward' effect (related to sine), this formula helps you find the combined effect without needing to measure the total 75-degree shot directly. It's like calculating the total distance covered by two auto-rickshaw rides without measuring the full journey at once, knowing only the individual parts.

Worked Example

Step-by-Step

Let's understand the geometric idea for proving sin (A+B) = sin A cos B + cos A sin B.

Step 1: Draw a line segment OP of length 'r' from the origin O in a coordinate plane. Let this line make an angle (A+B) with the positive x-axis.

Step 2: From point P, draw a perpendicular PQ to the x-axis. So, in triangle OQP, sin(A+B) = PQ/OP.

Step 3: Now, draw another line segment OR such that it makes an angle A with the positive x-axis. This line OR is inside the angle (A+B).

Step 4: From P, draw a perpendicular PS to the line OR. Also, from S, draw a perpendicular ST to the x-axis and a perpendicular SR' to PQ.

Step 5: Observe the triangles formed. In triangle OPS, sin A = PS/OP and cos A = OS/OP. Also, in triangle PSR', angle SPR' = angle ROS = A (alternate interior angles, if you extend PS to meet the x-axis).

Step 6: We know PQ = PR' + R'Q. Since PR' = ST (from rectangle STR'R), PQ = ST + R'Q. Also, R'Q = PS cos(angle SPR'). Since angle SPR' = A, R'Q = PS cos A.

Step 7: From triangle OST, ST = OS sin A. So, PQ = OS sin A + PS cos A.

Step 8: Substitute OS = OP cos A and PS = OP sin A (from triangle OPS) into the equation for PQ. We get PQ = (OP cos B) sin A + (OP sin B) cos A. (Here we consider angle POR as B, so in triangle OPS, angle SOP is B, and OS = OP cos B, PS = OP sin B).

Step 9: Divide the entire equation by OP to find sin(A+B). So, PQ/OP = (OP cos B sin A)/OP + (OP sin B cos A)/OP. This simplifies to sin(A+B) = sin A cos B + cos A sin B.

Answer: The proof geometrically shows how the components of sin(A+B) are built from sin A, cos A, sin B, and cos B.

Why It Matters

Understanding this proof is crucial for advanced mathematics, especially in calculus and physics, where combining wave functions or forces is common. Engineers use it to design sound systems and communication networks, while scientists in AI/ML apply similar trigonometric principles in signal processing and data analysis. It's a foundational concept for careers in sound engineering, telecommunications, and even space technology.

Common Mistakes

MISTAKE: Thinking sin(A+B) is simply sin A + sin B. | CORRECTION: Trigonometric functions don't distribute over addition like regular numbers. The formula sin(A+B) = sin A cos B + cos A sin B is specific and must be used.

MISTAKE: Confusing the sin(A+B) formula with cos(A+B). | CORRECTION: Remember that sin(A+B) involves a mix of sine and cosine terms with a '+' sign, while cos(A+B) involves cosine terms together, sine terms together, and a '-' sign.

MISTAKE: Incorrectly identifying angles or sides in the geometric proof diagram. | CORRECTION: Always label your diagram clearly. Ensure you correctly identify which angle is A and which is B, and which sides correspond to sine and cosine in each right-angled triangle.

Practice Questions

Try It Yourself

QUESTION: If A = 30 degrees and B = 60 degrees, verify sin(A+B) using the formula and compare it with sin(90 degrees). | ANSWER: sin(30+60) = sin 30 cos 60 + cos 30 sin 60 = (1/2)*(1/2) + (sqrt(3)/2)*(sqrt(3)/2) = 1/4 + 3/4 = 1. And sin(90) = 1. So, it matches.

QUESTION: Can the formula sin(A+B) = sin A cos B + cos A sin B be used if A or B are negative angles? Explain briefly. | ANSWER: Yes, the formula holds true for negative angles as well. Trigonometric identities are general and work for all real angles, positive or negative.

QUESTION: A radio signal's strength can be modeled using trigonometric functions. If two signals are combined, represented by sin(x) and cos(y), how would understanding sin(A+B) help in analyzing their combined effect if the total phase is (x+y)? | ANSWER: The sin(A+B) formula helps break down the combined signal sin(x+y) into components involving sin x, cos x, sin y, and cos y. This allows engineers to predict the amplitude and phase of the resulting signal, which is critical for clear communication.

MCQ

Quick Quiz

Which of the following is the correct expansion for sin(X + Y)?

sin X sin Y + cos X cos Y

sin X cos Y + cos X sin Y

sin X cos Y - cos X sin Y

cos X cos Y - sin X sin Y

The Correct Answer Is:

The correct formula for the sine of the sum of two angles is sin X cos Y + cos X sin Y. Options A and D are variations of the cosine formula, and C is the formula for sin(X-Y).

Real World Connection

In the Real World

In India, ISRO scientists use these trigonometric identities when calculating satellite orbits and trajectories. When a satellite adjusts its path, the change in its angle and position can be broken down using these formulas to precisely predict where it will be. Similarly, mobile network engineers use this to optimize signal strength and minimize interference when combining different frequency waves.

Key Vocabulary

Key Terms

Angle: The space between two intersecting lines or surfaces at or close to the point where they meet. | Sine: A trigonometric function of an angle, represented in a right-angled triangle as the ratio of the length of the side opposite the angle to the length of the hypotenuse. | Cosine: A trigonometric function of an angle, represented in a right-angled triangle as the ratio of the length of the adjacent side to the length of the hypotenuse. | Geometric Proof: A method of demonstrating a mathematical statement using shapes, points, lines, and angles.

What's Next

What to Learn Next

Now that you've understood the proof for sin(A+B), you should explore the proof for cos(A+B) = cos A cos B - sin A sin B. This builds directly on similar geometric principles and will complete your understanding of sum identities in trigonometry, which are super important for solving more complex problems!