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Trigonometry, EXT1 T3 EQ-Bank 25

Find all values of  `theta`  that satisfy the equation  `sqrt3 cos theta = sin(2theta)`.   (3 marks)

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`theta = 90° + m xx 180°, \ 60° + m xx 360°\ or\ 120° + m xx 360°`

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`sqrt3 cos theta` `= sin(2 theta)`
`2 cos theta sin theta-sqrt3 cos theta` `= 0`
`cos theta(sin theta-sqrt3/2)` `= 0`

COMMENT: Expressing “all values” is specifically mentioned in Topic Guidance as an application of arithmetic sequences.

`cos theta = 0 \or \  sin theta = sqrt3/2`

`theta = 90°, 270°\ or\ theta = 60°, 120° \ \ \ (0°<=theta<=360°)`

`:. theta = 90° + m xx 180°, \ 60° + m xx 360°\ or\ 120° + m xx 360°`

`text(where)\ m\ text(is an arbitrary integer).`

Trigonometry, EXT1 T3 EQ-Bank 23

Find all solutions of  `tan(2theta) = -tan theta`  for  `0<= theta<=2pi`.   (3 marks)

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`theta =0, \ pi/3, \ (2pi)/3, \ pi, \ (4pi)/3, \ (5pi)/3, \ 2pi`

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`(2 tan theta)/(1-tan^2 theta) = -tan theta`

`text(Let)\ \ tan theta = k.`

`(2k)/(1-k^2)` `= -k`
`2k` `= -k(1-k^2)`
`3k-k^3` `=0`
`k(3-k^2)` `= 0`

  
`k = 0, \ k = +- sqrt3`

`text(If)\ \  tan theta = 0 \ => \ theta=0, \ pi, \ 2pi`

`text(If)\ \ tan theta = +- sqrt 3 => \ theta = pi/3, \ (2pi)/3, \ (4pi)/3, \ (5pi)/3`

`:. theta =0, \ pi/3, \ (2pi)/3, \ pi, \ (4pi)/3, \ (5pi)/3, \ 2pi`

Trigonometry, EXT1 T3 EQ-Bank 21

 Solve  `sin(2x) = sin x`, for  `0<= x <=2 pi.`   (3 marks)

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`x = 0, \ pi/3, \ pi, \ (5pi)/3, \2 pi`

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`2sin x cos x = sin x`

MARKER’S COMMENT: Many students expanded `sin(2x)` and then cancelled `sin x` on both sides which lost a set of solutions!

`sinx(2cos x-1) = 0`

`sinx = 0, cosx = 1/2`

`text(If)\ \ sinx=0 \ => \ x=0, \ pi, \ 2pi`

`text(If)\ \ cos x=1/2 \ => \ x = pi/3, \ (5pi)/3`

`:. x = 0, pi/3, pi, (5pi)/3, 2pi`

Trigonometry, EXT1 T3 EQ-Bank 12

Given that  `cot(2x) + 1/2 tan(x) = a cot(x)`, calculate `a`.   (3 marks)

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`a = 1/2`

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`1/(tan(2x)) + (tan(x))/2 = a/(tan(x)),\ \ tan(x) != 0`

`(1-tan^2(x))/(2 tan(x)) + (tan(x))/2 = a/(tan(x))`

 
`text(If)\ \ tan(x) != 0 :`

`(1-tan^2(x)+tan^2(x))/(2tan(x))` `=a/(tan(x))`
`1` `=2a`
`:. a` `=1/2`

Trigonometry, EXT1 T3 EQ-Bank 22

Solve for `x`, given that

`x sin(x) sec(2x) = 0,\ \ 0<=x<=2pi`   (2 marks)

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`x = 0, \ pi\ \ text(or)\ \ 2pi`

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`xsin(x)sec(2x)= (xsin(x))/(cos(2x))=0`

`text(Find)\ x\ text(that satisfies:)`

`x = 0\ \ text(and)\ \ cos(2x) != 0 \ => \ x=0`

`text(or,)`

`sin(x) = 0\ \ text(and)\ \ cos(2x) != 0 \ => \ x=pi, \ 2pi`

`:. x = 0, \ pi\ \ text(or)\ \ 2pi`

Trigonometry, EXT1 T3 2009 HSC 3c

  1. Prove that  `tan^2 theta = (1-cos 2 theta)/(1 + cos 2 theta)`  provided that  `cos 2theta != -1`.   (2 marks)

  2. Hence find the exact value of `tan\ pi/8`.   (1 mark)

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a.    `text(Proof)\ \ text{(See Worked Solutions)}`

b.    `sqrt 2-1`

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a.    `text(Prove)\ \ tan^2 theta = (1-cos 2 theta)/(1 + cos 2 theta),\ cos 2 theta != -1`

`text(Using)\ \ cos 2 theta= 2 cos^2 theta-1= 1-2 sin^2 theta`

`text(RHS)` `= (1-(1-2 sin^2 theta))/(1 + (2 cos^2 theta-1))`
  `= (2 sin^2 theta)/(2 cos^2 theta)`
  `= tan^2 theta\ text(… as required.)`

 

b.    `tan^2\ pi/8` `= (1-cos (2 xx pi/8))/(1 + cos (2 xx pi/8))`
    `= (1-1/sqrt2)/(1 + 1/sqrt2) xx sqrt2/sqrt2`
    `= (sqrt2-1)/(sqrt2 + 1) xx (sqrt 2-1)/(sqrt2-1)`
    `= (sqrt 2-1)^2/(2-1)`
    `= (sqrt 2 -1)^2`

 
`:.\ tan\ pi/8 = sqrt 2-1\ \ \ \ \ (tan(pi/8)>0)`

`text{(}tan\ pi/8 = sqrt (3-2 sqrt 2)\ \ text{is also a correct answer)}`