Final answer:
To find the equation of the tangent line to a polar curve, we need to determine the slope of the tangent line at a given point and use the point-slope form. For the polar curve y = 1/2x, the equation of the tangent line is y - r*sin(theta) = (1/2)*cos(theta) + (1/2)*sin(theta)(x - r*cos(theta)).
Step-by-step explanation:
To find the equation of the tangent line to a polar curve, we need to determine the slope of the tangent line at a given point. The slope of the tangent line is given by the derivative of the polar equation with respect to theta (dθ/dt). To find the equation of the tangent line, we can use the point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the curve.
For the polar curve y = 1/2x, the derivative with respect to theta is given by dy/dtheta = (dy/dt)/(dx/dt). Since x = r*cos(theta) and y = r*sin(theta), we can substitute these expressions and simplify to find dy/dtheta = (1/2)*cos(theta) + (1/2)*sin(theta).
Therefore, the equation of the tangent line to the polar curve y = 1/2x at a given point (r, theta) is y - r*sin(theta) = (1/2)*cos(theta) + (1/2)*sin(theta)(x - r*cos(theta)).