The unit circle is an important idea in trigonometry. It helps us understand trigonometric functions better.

What is the Unit Circle?

The unit circle is a simple circle. It has a radius of 1 and is centered at the starting point (or origin) on a coordinate plane. Here are some important points about the unit circle:

1. What is It?: The unit circle can be described with the equation $x^2 + y^2 = 1$ This means that any point $(x, y)$ on the circle will fit this equation.

2. Coordinates: The points on the unit circle are linked to the cosine and sine of an angle $\theta$.

   • The x-coordinate is: $x = \cos(\theta)$
   • The y-coordinate is: $y = \sin(\theta)$ So, any point on the circle can be written as $(\cos(\theta), \sin(\theta))$.

3. Angles: Angles can be measured in either radians or degrees. Understanding the connection between angles and the unit circle is really important.

   • A full turn around the circle is equal to $360^\circ$ or $2\pi$ radians.
   • Some key angles in radians are: $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$.

4. Special Points: Certain angles give us specific points on the unit circle:

   • At $0^\circ$: $(1, 0)$
   • At $90^\circ$: $(0, 1)$
   • At $180^\circ$: $(-1, 0)$
   • At $270^\circ$: $(0, -1)$

5. Quadrants: The unit circle is divided into four parts called quadrants:

   • Quadrant I: $0 \leq \theta < \frac{\pi}{2}$, where both sine and cosine are positive.
   • Quadrant II: $\frac{\pi}{2} < \theta < \pi$, where sine is positive and cosine is negative.
   • Quadrant III: $\pi < \theta < \frac{3\pi}{2}$, where both sine and cosine are negative.
   • Quadrant IV: $\frac{3\pi}{2} < \theta < 2\pi$, where sine is negative and cosine is positive.

6. Finding Values: The unit circle helps us find the exact values of sine and cosine for common angles. For example:

   • $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$
   • $\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
   • $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$

7. Connection to Trigonometric Functions: The unit circle shows us how sine and cosine work.

   • Both sine and cosine repeat every $2\pi$.
   • You can find all trigonometric functions based on the relationships shown in the unit circle.

Knowing about the unit circle is key to understanding trigonometric functions, especially in Grade 12 Pre-Calculus.