The unit circle definition of trigonometric function

TL;DR

The unit circle extends the traditional SOH-CAH-TOA definitions of trigonometric functions and allows us to solve for angles beyond 90 degrees.

Transcript

We're now going to study the unit circle a little bit more, and see how it extends, I guess we could say, the traditional SOH-CAH-TOA definitions of functions. And how we can actually use it to solve for angles that the SOH-CAH-TOA definition of the trig functions actually doesn't help us with. So let's just, as a review, remember what SOH-CAH-TOA ... Read More

Key Insights

• 🍞 The unit circle extends the traditional SOH-CAH-TOA definitions of trigonometric functions.
• ❣️ The x-coordinate in the unit circle represents the cosine, while the y-coordinate represents the sine.
• ⭕ The unit circle provides a geometric understanding of trigonometric functions beyond 90 degrees.
• 🔺 Negative angles and angles greater than 90 degrees can be handled using the unit circle.
• ⭕ The unit circle allows for the calculation of trigonometric values for any angle on the circle.
• 👨‍💼 The tangent function can be determined by dividing the sine by the cosine of an angle.

Questions & Answers

Q: How does the unit circle extend the traditional definition of trigonometric functions?

The unit circle allows us to define sine and cosine for angles beyond 90 degrees, which is not possible with SOH-CAH-TOA. It provides a geometric understanding of these functions and their relationship to the coordinates of the unit circle.

Q: What is the significance of the x-coordinate and y-coordinate in the unit circle?

The x-coordinate represents the cosine of an angle, and the y-coordinate represents the sine of an angle. This relationship allows us to calculate the values of the sine and cosine functions for any angle on the unit circle.

Q: How can the unit circle be used to solve for angles greater than 90 degrees?

By understanding the relationship between angles on the unit circle and their corresponding coordinates, we can determine the sine and cosine values for angles greater than 90 degrees. This helps in solving trigonometric equations and problems involving such angles.

Q: How does the unit circle handle negative angles?

Negative angles in the unit circle can be treated as angles going in the opposite direction. For example, -30 degrees is equivalent to 330 degrees. The unit circle allows us to calculate the sine and cosine values for both positive and negative angles.

Summary & Key Takeaways

• SOH-CAH-TOA defines the sine, cosine, and tangent functions for right angles, but it becomes limited for angles beyond 90 degrees.

• The unit circle, a circle centered at (0,0) with a radius of 1, provides an extended definition of trigonometric functions.

• In the unit circle, the x-coordinate represents the cosine of an angle, and the y-coordinate represents the sine of an angle.