1. **Problem Statement:** Prove that on the unit circle, $\sin \theta = y$ and $\cos \theta = x$ where $(x,y)$ is a point on the circle corresponding to angle $\theta$. 2. **Definition of Unit Circle:** The unit circle is a circle centered at the origin $(0,0)$ with radius 1. Its equation is: $$x^2 + y^2 = 1$$ 3. **Coordinates on the Unit Circle:** Any point $(x,y)$ on the unit circle can be represented by an angle $\theta$ measured from the positive x-axis to the radius line connecting the origin to the point. 4. **Using Right Triangle Trigonometry:** Draw a right triangle by dropping a perpendicular from the point $(x,y)$ to the x-axis. The length of the adjacent side to angle $\theta$ is $x$, and the length of the opposite side is $y$. The hypotenuse is the radius of the circle, which is 1. 5. **Trigonometric Ratios:** By definition of sine and cosine in a right triangle: $$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{1} = y$$ $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{1} = x$$ 6. **Conclusion:** Thus, for any angle $\theta$ on the unit circle, the coordinates of the point are: $$(x,y) = (\cos \theta, \sin \theta)$$ which proves the statement.