An object is moving around the unit circle with parametric equations x(t)=cos(t)

Question

An object is moving around the unit circle with parametric equations x(t)=cos(t), y(t)=sin(t), so it's location at time t is P(t)=(cos(t),sin(t)) . Assume 0 < t < /2. At a given time t, the tangent line to the unit circle at the position P(t) will determine a right triangle in the first quadrant. (Connect the origin with the y-intercept and x-intercept of the tangent line.)

The identity

sin(2t)=2sin(t)cost(t)

might be useful in some parts of this question.

(e) With our restriction on t, the smallest t so that a(t)=2 is  

(f) With our restriction on t, the largest t so that a(t)=2 is  

(g) The average rate of change of the area of the triangle on the time interval [/6,/4] is  .

(g) The average rate of change of the area of the triangle on the time interval [/4,/3] is  .

Explanation / Answer

P(t) = <cos(t), sin(t)>
P'(t) = <-sin(t), cos(t)>

P(t) defines the position and P'(t) defines the direction. A vector for the tangent line is:
R(t) = <cos(t), sin(t)> + k<-sin(t), cos(t)>
Where k is an arbitrary constant.

we want to know when the vector touches the x-axis:
cos(t) - ksin(t) = 0
k = cot(t)
And the y value of this is:
y = sin(t) + kcos(t)
y = sin(t) + cos(t)cot(t) = 1/sin(t)

now we want to know when the vector touches the y-axis:
sin(t) + kcos(t) = 0
k = -tan(t)
And the x value of this is:
x = cos(t) - ksin(t)
x = cos(t) + sin(t)tan(t) = 1/cos(t)

The area of a triangle is:
A = bh / 2

b = 1/cos(t) and h = 1/sin(t)

=> A = 1/2cos(t)sin(t) = 1/sin(2t)

now A = 2 = 1/sin(2t)

sin(2t) = 1/2

when 2t = pi/6

or t = pi/12

so when t = pi/12 we'll get the area as 2

g>

average rate of change over [pi/6 , pi/4] is = [a(pi/4) - a(pi/6)]/[pi/4 - pi/6] = [1 - 2/sqrt3]/(pi/12)

={12* [sqrt(3) - 2]}/sqrt(3)pi

h> average rate of change over [pi/4 , pi/3] is = [a(pi/3) - a(pi/4)]/[pi/3 - pi/4] = [2/sqrt(3) - 1]/(pi/12)

= {12*[2-sqrt(3)]}/sqrt(3)pi

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.