# How do you find the horizontal and vertical tangents to #x = Cos(3t)# and #y = 2sin(t)#?

Recall that

Therefore

Vertical tangents occur when the derivative is undefined.

Hopefully this helps!

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Horizontal Tangents occur when

Vertical Tangents occur when

We have the following parametric equation:

# x = cos(3t) # ... [A]

# y =2sin(t) # ... [B]The entire curve is mapped out for

#tin[0,2pi]# .Horizontal tangents occurs when

#dy/dt=0# , and vertical tangents occur when#dx/dt=0# , This arrises from the chain rule, as:

# dy/dx = (dy//dt) /(dx//dt)# Horizontal Tangents:

Differentiating [B] wrt

#t# we get:

# dy/dt = 2cost #

# dy/dt = 0 => 2cost = 0 #

# :. cost=0 => t = pi/2, (3pi)/2 \ " for " \ t in [0,2pi]# Vertical Tangents:

Differentiating [A] wrt

#t# we get:

# dx/dt = -3sin(3t) #

# dx/dt = 0 => -3sin(3t) = 0 #

# :. sin(3t)=0 => 3t = 0, pi, 2pi, 3pi, 4pi, 5pi, 6pi \ " for " \ 3t in [0,6pi]#

# :. t = 0, pi/3, (2pi)/3, pi, (4pi)/3, (5pi)/3, 2pi \ " for " \ t in [0,2pi]# The last value of

#t# also corresponds to#t=0# , so can omit this value

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To find the horizontal and vertical tangents, we need to find the derivative of the given equations with respect to t.

For x = cos(3t), the derivative is dx/dt = -3sin(3t).

For y = 2sin(t), the derivative is dy/dt = 2cos(t).

To find the horizontal tangents, we set dy/dt = 0 and solve for t.

2cos(t) = 0 cos(t) = 0 t = π/2, 3π/2

To find the vertical tangents, we set dx/dt = 0 and solve for t.

-3sin(3t) = 0 sin(3t) = 0 3t = 0, π, 2π t = 0, π/3, 2π/3

Therefore, the horizontal tangents occur at t = π/2 and 3π/2, and the vertical tangents occur at t = 0, π/3, and 2π/3.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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