Introduction to the unit circle | Trigonometry | Khan Academy | Summary and Q&A

TL;DR

This video explains how to use the unit circle to extend our traditional definitions of trigonometric functions.

Key Insights

• 🔨 The unit circle is a useful tool for understanding trigonometry.
• 😥 By using the coordinates of points on the unit circle, we can extend the definitions of trigonometric functions.
• ❣️ The x-coordinate represents the cosine of the angle, while the y-coordinate represents the sine.
• 💦 The unit circle helps us work with angles beyond 90 degrees, where traditional definitions of trigonometric functions break down.
• 🥳 Tangent can be defined as the ratio of sine to cosine.
• 🔺 The unit circle allows us to evaluate trigonometric functions for any angle, not just right triangles.
• 🔄 Positive angles are measured counterclockwise, while negative angles are measured clockwise.

Transcript

What I have attempted to draw here is a unit circle. And the fact I'm calling it a unit circle means it has a radius of 1. So this length from the center-- and I centered it at the origin-- this length, from the center to any point on the circle, is of length 1. So what would this coordinate be right over there, right where it intersects along the ... Read More

Questions & Answers

Q: What is the unit circle and how does it relate to trigonometry?

The unit circle is a circle with a radius of 1 centered at the origin. It helps us visualize coordinates and angles in trigonometry.

Q: How do we determine the coordinates of points on the unit circle?

The x-coordinate of a point on the unit circle is equal to the cosine of the angle, while the y-coordinate is equal to the sine of the angle.

Q: How does the unit circle help us extend the definitions of trigonometric functions?

By using the coordinates of points on the unit circle, we can define cosine as the x-coordinate and sine as the y-coordinate. This allows us to evaluate trigonometric functions for any angle, not just right triangles.

Q: What is the relationship between the unit circle and positive angles?

Positive angles are measured counterclockwise from the positive x-axis. The terminal side of an angle intersects the unit circle at a point whose coordinates represent the values of cosine and sine.

Summary & Key Takeaways

• The unit circle is a circle with a radius of 1 and is centered at the origin. It helps us visualize coordinates and angles.

• Positive angles are measured counterclockwise from the positive x-axis, while negative angles are measured clockwise.

• The sine of an angle is equal to the y-coordinate of the point where the angle intersects the unit circle, and the cosine is equal to the x-coordinate.

• By using the unit circle, we can extend the definitions of trigonometric functions beyond right triangles.