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Trigonometry, 2ADV T2 2004 HSC 9a

Consider the geometric series  `1-tan^2 theta + tan^4 theta- …`

  1. When the limiting sum exists, find its value in simplest form.   (2 marks)

  2. For what values of  `theta`  in the interval
  3. `-pi/2 < theta < pi/2`  does the limiting sum of the series exist?   (2 marks)

Show Answers Only

a.    `cos^2 theta`

b.    `-pi/4 < theta < pi/4`

Show Worked Solution

a.    `1-tan^2 theta + tan^4 theta- …`

`=>\ text(GP where)\ \ a=1,\ \ r=T_2/T_1= − tan^2 theta`

`:. S_∞` `= 1/(1-(-tan^2 theta))`
  `= 1/(1 + tan^2 theta)`
  `= 1/(sec^2 theta)`
  `= cos^2 theta`

  
b.    `text(Find)\ theta\ text(such that)\ |r |<1:`

`|-tan^2 theta\ |` `< 1`
` tan^2 theta` `< 1`
`-1 < tan theta` `< 1`
`:. -pi/4 < theta` `< pi/4`