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

Question

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

Isaiah Allen · Accepted Answer

Recall that

#(dy/(dt))/(dx/(dt)) = dy/dx#

Therefore

#dy/dx = (2cost)/(-3sin(3t))#

Horizontal tangents occur when the derivative equals #0#.

#0 = 2cost -> t = pi/2 + pin#

Vertical tangents occur when the derivative is undefined.

#-3sin(3t) =0 -> 3t = pin -> t = pi/3n#

Hopefully this helps!

Samantha Adkison · Answer

Horizontal Tangents occur when #t = pi/2, (3pi)/2 #

Vertical Tangents occur when #t = 0, pi/3, (2pi)/3, pi, (4pi)/3, (5pi)/3#

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

William Abdalla · Answer

undefined 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.