What Is the Angle Addition Formula for Cosine?

TL;DR

The angle addition formula for cosine states that cos(X + Y) equals cos(X) * cos(Y) - sin(X) * sin(Y). This formula is proven using right triangles and trigonometric ratios, demonstrating that the lengths of specific segments can be used to derive the relationship between the angles.

Transcript

Voiceover: In the last video we proved the angle addition formula for sine. You could imagine in this video I would like to prove the angle addition for cosine, or in particular, that the cosine of X plus Y, of X plus Y, is equal to the cosine of X. Cosine of X, cosine of Y, cosine of Y minus, so if we have a plus here we're going to have a minus h... Read More

Key Insights

• 🎮 The video provides a visual approach to understanding the angle addition formula for cosine.
• 😒 It highlights the use of right triangles and trigonometric ratios in the proof.
• 👨‍💼 The proof demonstrates the derivation of segment AB as cosine X times cosine Y and segment FB as sine X times sine Y.

Questions & Answers

Q: How is cosine of X plus Y expressed in terms of segments in the video?

Cosine of X plus Y is represented as the length of segment AF, which is equivalent to segment AB minus segment FB.

Q: How does the proof establish that segment AB is equal to cosine X times cosine Y?

By considering right triangle ACB, it is shown that segment AB, adjacent to angle Y, is equal to cosine of X times cosine of Y.

Q: How is segment FB defined in the proof?

Since ECBF is a rectangle, segment FB is equivalent to segment EC. By applying sine of Y, it is proven that segment FB is equal to sine X times sine Y.

Q: What is the final expression for cosine of X plus Y in the proof?

Cosine of X plus Y is equal to cosine X times cosine Y minus sine X times sine Y.

Summary & Key Takeaways

• The video aims to prove the angle addition formula for cosine.

• It uses triangles and trigonometric ratios to express cosine of X plus Y as the difference of two segments.

• The proof shows that segment AB is equal to cosine X times cosine Y, and segment FB is equal to sine X times sine Y.

• By subtracting FB from AB, the video concludes that cosine of X plus Y is equal to cosine X times cosine Y minus sine X times sine Y.