Pure: Straight Line

A. Trigonometric Ratios in the Cartesian Plane Definition: Trigonometric ratios (sine, cosine, tangent) are defined for any angle using a point P(x,y) on the terminal arm of the angle, with 'r' being the distance from the origin to P. Explanation: Instead of being limited to right-angled triangles, these ratios extend to all angles by placing the angle's vertex at the origin and its initial arm along the positive x-axis. If P(x,y) is any point on the terminal arm, and r = sqrt(x² + y²) is the distance from the origin to P, then sin(alpha) = y/r, cos(alpha) = x/r, and tan(alpha) = y/x. This allows us to define trigonometry for angles greater than 90 degrees or negative angles. Think of it as: Imagine a rotating arm (radius 'r') sweeping around a clock face. The 'x' and 'y' coordinates of the arm's tip at any moment define its position, and these coordinates, relative to the arm's length, give you the sine and cosine values, like how a shadow changes length as the sun moves.