Question

Find the parametric equations for the tangent line to the curve with the given parametric equations at the specified point. The parametric equations are x = 6 cos(t), y = 6 sin(t), z = 6 cos(2t). Find the tangent line at the point (3, 3, 3).

Answer 1

To find the parametric equations for the tangent line to the curve at the specified point, we need to find the derivative of the given parametric equations and substitute t = 25 into these derivatives. The parametric equations for the tangent line are x = 6cos(25) + (-6sin(25))(t - 25), y = 6sin(25) + (6cos(25))(t - 25), and z = 6cos(50) + (-12sin(50))(t - 25).

Step-by-step explanation:

To find the parametric equations for the tangent line to the curve at the specified point, we need to find the derivative of the given parametric equations. Derivative of x with respect to t is -6sin(t), derivative of y with respect to t is 6cos(t), and derivative of z with respect to t is -12sin(2t). Substituting t = 25 into these derivatives, we get the slope of the tangent line. So, the parametric equations for the tangent line are x = 6cos(25) + (-6sin(25))(t - 25), y = 6sin(25) + (6cos(25))(t - 25), and z = 6cos(50) + (-12sin(50))(t - 25).