Definition
The unit circle is a circle with a radius of $1$ centered at the origin $(0,0)$ in the Cartesian coordinate plane. Its equation is: %%MATH_DISPLAY_0%% In trigonometry, the unit circle is used \to define the sine and cosine functions for all real angles. For an angle $theta$ measured from the positive $x$-axis, the coordinates $(x, y)$ of the point where the terminal side of the angle intersects the unit circle are: %%MATH_DISPLAY_1%% ## Worked Example **Find $sinleft(frac{pi}{4}right)$ and $cosleft(frac{pi}{4}right)$ using the unit circle.** 1. The angle $frac{pi}{4}$ radians ($45^circ$) is measured from the positive $x$-axis. 2. On the unit circle, the coordinates at this angle are: %%MATH_DISPLAY_2%% 3. For $frac{pi}{4}$, both $x$ and $y$ are equal, and since $x^2 + y^2 = 1$: %%MATH_DISPLAY_3%% 4. Therefore, %%MATH_DISPLAY_4%% ## Key Takeaways - The unit circle provides a geometric definition of sine and cosine for all angles. - Any point $(x, y)$ on the unit circle corresponds \to $(costheta, sintheta)$ for some angle $theta$. - The unit circle helps visualize and compute trigonometric values beyond $0^circ$ \to $360^circ$ (or $0$ \to $2pi$ radians).