Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x = cos(θ) + sin(10θ) y = sin(θ) + cos(10θ) θ = 0 y(x) =

The equation of the tangent to the curve at the point corresponding to the given values of the parametric equations given is;

y - 1 = ¹/₁₀(x - 1)

We are given;

x = cos θ + sin(10θ)

y = sin θ + cos(10θ)

Since we want to find equation of tangent, let us first differentiate with respect to θ. Thus;

dx/dθ = -sin θ + 10cos (10θ)

Similarly;

dy/dθ = cos θ - 10sin(10θ)

To get the tangent dy/dx, we will divide dy/dθ by dx/dθ to get;

(dy/dθ)/(dx/dθ) = dy/dx =  (cos θ - 10sin(10θ))/(-sin θ + 10cos(10θ))

To get the tangent, we will put the angle to be equal to zero.

Thus, at θ = 0, we have;

dy/dx = (cos 0 - 10sin 0)/(-sin 0 + 10cos 0)

dy/dx = 1/10

Also, at θ = 0, we can get the x-value and y-value of the parametric functions.

Thus;

x = cos 0 + sin 0

x = 1 + 0

x = 1

y = sin 0 + cos 0

y = 0 + 1

y = 1

Thus, the equation of the tangent line to the curve in point slope form gives us;

y - 1 = ¹/₁₀(x - 1)

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Step-by-step explanation:

x = cos θ + sin(10θ)

y = sin θ + cos(10θ)

Take derivative with respect to θ:

dx/dθ = -sin θ + 10 cos(10θ)

dy/dθ = cos θ - 10 sin(10θ)

Divide:

dy/dx = (dy/dθ) / (dx/dθ)

dy/dx = (cos θ - 10 sin(10θ)) / (-sin θ + 10 cos(10θ))

Evaluate the derivative at θ=0:

dy/dx = (cos 0 - 10 sin 0) / (-sin 0 + 10 cos 0)

dy/dx = 1/10

Evaluate the parametric functions at θ=0:

x = cos 0 + sin 0 = 1

y = sin 0 + cos 0 = 1

Writing the equation of the tangent line in point-slope form:

y - 1 = 1/10 (x - 1)

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step-by-step explanation:

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