Find the equation of the tangent line to the curve at the given value of t:
x=sin(t); y=(cos(t))^2; t=(pi/3)
x = sin t
dx/dt = cost
y = cos^2 t
dy/dt = 2cost(-sint) = -2sint cost
slope = dy/dx = (dy/dt) / (dx/dt
= cost/(-2sintcost) = -1/(2sint)
when t = π/3
dy/dt = -1/2sin(π/3) = -1/(2√3/2) = -1/√3
when t = π/3 , x = sin π/3 = √3/2
and y = cos^2 (π/3) = 1/4
so we have a point (√3/2 , 1/4) with slope √3/2
y - 1/4 = (√3/2)(x-√3/2)
2y - 1/2 = √3x - 3/4
√3x - 2y = 1/4
To find the equation of the tangent line to a curve at a given value of t, you need to first find the derivative of the curve with respect to t and then substitute the given value of t into the derivative to find the slope of the curve at that point. Finally, you can use the slope and the point to write the equation of the tangent line using the point-slope form.
Let's start by finding the derivative of the curve with respect to t:
Given:
x = sin(t)
y = (cos(t))^2
To find dx/dt, we can apply the chain rule:
dx/dt = d(sin(t))/dt = cos(t)
To find dy/dt, we can differentiate each term separately using the chain rule:
dy/dt = d((cos(t))^2)/dt = 2(cos(t))*(-sin(t)) = -2sin(t)*cos(t)
Now, let's substitute t = pi/3 into the derivatives to find the slope of the curve at that point:
dx/dt at t = pi/3: cos(pi/3) = 1/2
dy/dt at t = pi/3: -2sin(pi/3)*cos(pi/3) = -2(√3/2)(1/2) = -√3
So, the slope of the curve at t = pi/3 is -√3.
Next, we need to find the coordinates of the point on the curve at t = pi/3.
Plugging t = pi/3 into the given equations:
x = sin(pi/3) = √3/2
y = (cos(pi/3))^2 = (1/2)^2 = 1/4
Therefore, the point on the curve at t = pi/3 is (√3/2, 1/4).
Now we have both the slope (-√3) and a point (√3/2, 1/4) on the curve. We can use the point-slope form of the equation of a line to write the equation of the tangent line:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) is the given point on the line.
Plugging in the values, we get:
y - 1/4 = -√3(x - √3/2)
Simplifying the equation, we get the equation of the tangent line:
y = -√3x + 3/2 + 1/4
Therefore, the equation of the tangent line to the curve at t = pi/3 is:
y = -√3x + 7/4