Classic math proofs of the law of sine and the law of cosine

TL;DR

This video explains the law of sine and law of cosine for acute angles in triangles, providing step-by-step demonstrations and proofs.

Transcript

okay video will be proving both the law of sine  and the law of cosine and we'll just be focusing   on acute angles and the same for any angles in  triangle right anyway let's start with law of   sine first let me show you guys a picture right  here and I'm gonna label this to be capital a   capital B and capital C and this right here will  be just... Read More

Key Insights

• 👨‍💼 The law of sine states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant.
• 🔺 The law of cosine is a generalized version of the Pythagorean theorem that relates the sides and angles of a triangle.
• 👮 Different versions of the law of cosine can be used depending on which side of the triangle is being calculated.
• 🔺 The law of sine and law of cosine are useful for calculating missing sides and angles in a triangle for acute angles.
• 👍 The Pythagorean theorem is a fundamental tool used in proving the law of cosine.

Questions & Answers

Q: How is the law of sine derived?

The law of sine is derived by setting up a triangle, selecting a base side, and determining the sine of each angle in relation to the base side. By equating the two equations, sine(a)/a = sine(C)/C is obtained.

Q: Can the law of cosine be used without the Pythagorean theorem?

Yes, the law of cosine can be used without the Pythagorean theorem. However, in this video, the law of cosine is demonstrated using the Pythagorean theorem to show the relationship between the sides of a triangle.

Q: Are there different versions of the law of cosine?

Yes, there are different versions of the law of cosine depending on which side of the triangle is being calculated. In the video, the version C^2 = a^2 + b^2 - 2ab * cosine(C) is used to calculate side C.

Q: Can the law of sine and law of cosine be applied to any angle in a triangle?

The law of sine and law of cosine can be applied to any angle in a triangle, but in this video, the focus is on acute angles.

Summary & Key Takeaways

• The law of sine is demonstrated through a triangle, showing that H1 = C * sine(a) and H1 = a * sine(C), ultimately leading to the relationship of sine(a)/a = sine(C)/C.

• The law of cosine is explained with the use of the Pythagorean theorem, showing that C^2 = a^2 + b^2 - 2ab * cosine(C).

• Different versions of the law of cosine are mentioned, depending on which side of the triangle is being calculated.