Definition
The unit circle is a circle centered at the origin $(0,0)$ in the coordinate plane with a radius of $1$. For any angle $theta$ (measured from the positive $x$-axis), the coordinates of the point where the terminal side of the angle intersects the unit circle are $(costheta, sintheta)$. Thus: - $costheta$ is the $x$-coordinate. - $sintheta$ is the $y$-coordinate. Other trigonometric values can be found using these coordinates: - $tantheta = frac{sintheta}{costheta}$ --- ## Worked Example **Find $sin 45^circ$, $cos 45^circ$, and $tan 45^circ$ using the unit circle.** 1. **Convert \to radians:** %%MATH_DISPLAY_0%% 2. **Locate the point on the unit circle:** At $theta = frac{pi}{4}$, the coordinates are: %%MATH_DISPLAY_1%% 3. **Find the values:** - $sin 45^circ = frac{sqrt{2}}{2}$ - $cos 45^circ = frac{sqrt{2}}{2}$ - $tan 45^circ = frac{sin 45^circ}{cos 45^circ} = frac{frac{sqrt{2}}{2}}{frac{sqrt{2}}{2}} = 1$ --- ## Key Takeaways - The unit circle allows you \to find $sintheta$ and $costheta$ as the $y$ and $x$ coordinates, respectively.
- Trigonometric values for common angles can be read directly from the unit circle.
- The unit circle helps visualize and compute all six trigonometric functions for any angle.