Find the coordinates of the point of tangency, which is the point where the circle and the tangent line meet. Let the center of the circle be at the origin of the x-y coordinate plane, so the equation for the circle is x^2 + y^2 = R^2, where R is the radius of the circle. Enter the known coordinate into the equation and solve for the unknown. For example, suppose R = 2 and x = 1, the equation would be 1^2 + y^2 = 2^2. Solving for y results in two solutions, y = 1.73 and y = -1.73, because the line x = 1 intersects the circle in two places. Assuming that only the positive coordinates are desired, the tangent line comes into contact with the circle at (1, 1.73).
Calculate the slope of the diameter which has the point of tangency as one of its endpoints. The slope is the change in y divided by the change in x. Since the diameter passes through the origin, that simplifies to the y-coordinate divided by the x-coordinate. In the given example, slope = 1.73/1 = 1.73.
Find the slope of the tangent line. A tangent line to a circle is always perpendicular to the diameter whose endpoint is the point of tangency. Therefore, the slope of the tangent is the negated reciprocal of the slope of the diameter: slope of tangent = -1/(slope of diameter). For example, the slope of the tangent = -1/1.73 = -0.578.
Calculate the angle of the tangent line. Since the tangent of an angle equals the slope, find the arc-tangent of the slope. For example, angle = arctan(-0.578) = -30.0 degrees.