Determine the equation of the tangent line to the given path at the specified value of t. (Enter your answer as a comma-separated list of equations in (x, y, z) coordinates.) (sin(3t), cos(3t), 2t7/2); t=1

k(t) =  (sin3, cos3, 2) + [(t - 1)(cos3, -3sin3, 7)]

Step-by-step explanation:

For a path r(t), the general equation k(t) of its tangent line at a specified point r(t₀) is given by;

k(t) = r(t₀) + r'(t₀) [t - t₀]            -----------------(i)

Where

r'(t) is the first derivative of the path r(t) at a given value of t.

From the question:

r(t) =  (sin3t, cos3t, 2) and t₀ = 1

=> r(1) = (sin3, cos3, 2) at t₀ = 1

Find the first derivative component-wise of r(t) to get r'(t)

∴ r'(t) = (cos3t, -3sin3t, 7)

=>r'(1) = (cos3, -3sin3, 7)

Now, at t₀ = 1, equation (i) becomes;

k(t) = r(1) + [ r'(1) (t-1)]    [substitute the necessary values]

k(t) =  (sin3, cos3, 2) + [(t - 1)(cos3, -3sin3, 7)]

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step-by-step explanation: