What Is the Proof for sin(a + b) Formula?

TL;DR

The proof demonstrates that sin(a + b) equals cos(a)sin(b) + sin(a)cos(b). Using geometric relationships and trigonometric identities, the proof rewrites the sine of the sum of two angles entirely in terms of the individual sine and cosine functions of those angles, showcasing the identity's symmetry and elegance.

Transcript

Welcome back. I'm now going to do a proof of a trig identity, which I think is pretty amazing. Although, I think, the proof isn't that obvious. And I'll have to admit ahead of time, this isn't something that would have occurred to me naturally. I wouldn't have naturally drawn this figure just to start off with. Let's just say we want to figure out ... Read More

Key Insights

• 😒 The proof demonstrates the use of diagrams to visualize and derive trigonometric identities.
• 🍉 Rewriting lengths in terms of other segments allows for the manipulation and rearrangement of terms to simplify expressions.
• 💅 The symmetry of the final trigonometric identity highlights the elegance and beauty of its derivation.
• 🫥 The proof involves concepts of right triangles, parallel lines, and trigonometric ratios.
• 🔺 Understanding the relationships between angles and corresponding sides in various right triangles is crucial for the proof.
• 🥳 The choice of specific segments, such as BD/BE and EF/AE, enables the proof to relate them to sine and cosine ratios.
• 👍 The proof showcases the creativity and ingenuity required to discover and prove trigonometric identities.

Questions & Answers

Q: How does the proof start in finding an alternative representation for the sine of alpha plus beta?

The proof begins by considering a right triangle with angle alpha, angle beta, and the sum of these angles (alpha plus beta). It aims to determine the sine of the sum, which is opposite over hypotenuse for this combined angle.

Q: What is the significance of rewriting BC as the sum of BD and EF?

By rewriting BC as BD plus EF, the proof introduces new segments that can be further related to trigonometric ratios. This allows for the manipulation and rearrangement of terms to ultimately arrive at a simplified representation of the sine of alpha plus beta.

Q: Why does the proof choose specific segments such as BE and AE?

The proof selects segments like BE and AE because they correspond to sides in other right triangles. By relating these segments to trigonometric ratios (adjacent/hypotenuse and opposite/hypotenuse), the proof can express ratios such as BD/BE and EF/AE in terms of cosine and sine of the angles alpha and beta.

Q: What is the final trigonometric identity derived from the proof?

The final result of the proof shows that the sine of alpha plus beta is equal to the cosine of alpha times the sine of beta, plus the sine of alpha times the cosine of beta. This identity provides an alternative way to represent the sine of the sum of two angles.

Summary & Key Takeaways

• The proof aims to find an alternative way to express the sine of the sum of two angles, alpha and beta, in terms of the individual sines and cosines of those angles.

• By using a diagram and rewriting the lengths in terms of other segments, the proof arrives at the conclusion that the sine of alpha plus beta is equal to the cosine of alpha times the sine of beta, plus the sine of alpha times the cosine of beta.

• The proof demonstrates the symmetry of the trigonometric identity and showcases the elegance of its derivation.