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Figure 29 : Parametric Equations
Product no. Program 53 15
Q1. What is the graph of x = cos(t) and y = sin(t)
A1. It is a unit circle
  • x^2 + y^2 = cos^2(t) + sin(t)^2 = 1
  • It is circle with center at (0,0) and radius = 1
Q2. Find parametric equations of unit hyperbola
A2. The parametric equations are
  • x = sec(t) and y = tan(t)
  • x = tan(t) and y = sec(t)
  • x = csc(t) and y = cot(t)
  • x = cot(t) and y = csc(t)
Q3. Prove that x=cos(t) and y=sin(t) is a unit circle
A3. Proof
  • Use Pytnagorean relation cos(t)^2 + sin(t)^2 = 1
  • We have x^2 + y^2 = cos(t)^2 + sin(t)^2 = 1
  • x^2 + y^2 = 1 is a circle with center at (0,0) and radius = 1
Q4. Prove that x=tan(t) and y=ses(t) is a unit hyperbola
A4. Proof
  • Use Pytnagorean relation 1 + tan(t)^2 = sec(t)^2
  • We have x^2 - y^2 = tan(t)^2 - sec(t)^2 = -1
  • x^2 - y^2 = -1 is a unit hyperbola with semi-axis 1
Q5. Sketch x=tan(t) and y=sec(t) on internet
A5. Use Program ABG program 15 03
  • Run program ABG
  • Click start
  • Click section 15 in upper box
  • Click program 03 for x=tan(x)
  • Give a number 5 for y=sec(x)
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