Use the axis intercepts of the unit circle -- (1, 0), (0, 1), (-1, 0) and (0, -1) -- to determine the signs of points on the unit circle. Consider the order that the points are encountered when the line segment from (0, 0) to (1, 0) is swept counterclockwise around (0, 0). The first intercept encountered is (1, 0) and the next is (0, 1). All the signs of the points in this quarter are (+, +) as indicated by the intercepts. The points in the next quadrant are between (0, 1) and (-1, 0) so the signs will be (-, +) and so forth.
Find the geometric and algebraic relationships of the points on the unit circle by drawing a vertical line from a point to the x-axis. This vertical line is the height of a right triangle whose base is along the x-axis and whose hypotenuse is the radius of the unit circle. If you know the x-coordinate of a point on the unit circle, you can calculate the y-coordinate: Y = (1 - x^2)^0.5. Similarly, if you know the y-coordinate, the x-coordinate equals 1 - y^2)^0.5.
See the relationships between the trigonometric functions in a memorable way with the unit circle. The height of the triangle under a point on the unit circle is the sine of the angle associated with the point. The base of the triangle is the cosine of the angle. If you draw a tangent to the unit circle at (1, 0) and extend the radius from the origin to the place where it intersects the tangent line, you will have geometric representations of the secant and tangent functions. These mental images allow you to see what happens to the trigonometric functions as the angle increases.