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Trigonometry, SPEC2 2011 VCAA 9 MC

The number of distinct solutions of the equation

`xsin(x)sec(2x) = 0, \ x ∈ [0,2pi]`  is

  1. 3
  2. 4
  3. 5
  4. 6
  5. 7
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`A`

Show Worked Solution
`xsin(x)sec(2x)` `= (xsin(x))/(cos(2x))=0`

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

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

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

`=> A`

Trigonometry, SPEC1 2012 VCAA 2

Find all real solutions of the equation  `2 cos(x) = sqrt 3 cot (x).`  (3 marks)

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`x=((2n+1)pi)/2, pi/3 + 2npi, (2pi)/3 + 2npi\ \ \ (n in ZZ)`

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`2 cos x = sqrt 3 cot x`

`2 cos x – sqrt 3 cot x` `= 0`  
`2 cos x – sqrt 3 (cos x)/(sin x)` `=0`  
`(2 – sqrt 3/sin x) cos x` `=0`  

 
`text(Solution 1:)\ \ cosx=0`

♦♦ Mean mark 28%.

`x=pi/2, (3pi)/2, (5pi)/2, …`

`x=((2n+1)pi)/2\ \ \ (n in ZZ)`
 

`text(Solution 2:)\ \ 2 – sqrt 3/sin x = 0\ \ =>\ \ sin x = sqrt 3/2`

`x = pi/3, (2pi)/3, (7pi)/3, (8pi)/3, …`

`x=pi/3 + 2npi, (2pi)/3 + 2npi\ \ (n in ZZ)`

Trigonometry, SPEC1-NHT 2017 VCAA 6

Find all real solutions of  `tan(2x) = -tan(x)`.  (3 marks)

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`x = k pi, \ pi/3 + k pi, \ (2 pi)/3 + k pi, k in ZZ`

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`(2 tan (x))/(1 – tan^2(x)) = -tan(x)`

 
`text(Let)\ \ tan(x) = k,`

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

 
`k(3 – k^2) = 0 \ \ =>\ \  k = 0, k^2 = 3`

`=> tan(x) = 0, tan(x) = +- sqrt 3`

`:. x = k pi, \ pi/3 + k pi, \ (2 pi)/3 + k pi\ \ (k in ZZ)`

Trigonometry, SPEC1 2015 VCAA 7

  1. Solve \(\sin (2 x)=\sin (x), x \in[0,2 \pi]\).  (3 marks)

  2. Find \(\left\{x: \operatorname{cosec}(2 x)<\operatorname{cosec}(x), x \in\left(0, \dfrac{\pi}{2}\right) \cup\left(\dfrac{\pi}{2}, \pi\right)\right\}\)   (2 marks)

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  1. \(x=0, \pi, 2 \pi\) or \(x=\dfrac{\pi}{3}, \dfrac{5 \pi}{3}\)
  2. \(x \in\left(0, \dfrac{\pi}{3}\right) \cup\left(\dfrac{\pi}{2}, \pi\right)\)
Show Worked Solution

a.  \(2 \sin (x) \cos (x)=\sin (x)\)

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

\(\sin (x)(2 \cos (x)-1)=0\)

\(\sin (x)=0, \cos (x)=\dfrac{1}{2}\)

\(x=0, \dfrac{\pi}{3}, \pi, 2 \pi-\dfrac{\pi}{3}, 2 \pi\)

\(x=0, \dfrac{\pi}{3}, \pi, \dfrac{5 \pi}{3}, 2 \pi\)
 

b.  \(\text {Solving} \ \operatorname{cosec}(2 x)=\operatorname{cosec}(x), x \in(0, \pi)\left\{\dfrac{\pi}{2}\right\}:\)

♦♦ Mean mark 26%.

\(\text{Using part (a): }\)

\(\sin (2 x)=\sin (x)\) when \(x=\dfrac{\pi}{3}\)

\(\therefore \operatorname{cosec}(x)=\operatorname{cosec}(2 x)\) at \(x=\dfrac{\pi}{3}\)

\(\text {Sketch graphs:}\)
 

 

\(\text {When} \ \operatorname{cosec}(2 x)<\operatorname{cosec}(x):\)

\(x \in\left(0, \dfrac{\pi}{3}\right) \cup\left(\dfrac{\pi}{2}, \pi\right)\)

Trigonometry, SPEC2 2018 VCAA 4 MC

If  `cos(x) = -a`  and  `cot(x) = b`, where  `a, b > 0`, then  `text{cosec}(-x)`  is equal to

  1. `b/a`
  2. `-b/a`
  3. `-a/b`
  4. `a/b`
  5. `-ab` 
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`A`

Show Worked Solution

`cos(x) < 0, \ cot(x) > 0 => x\ text(is in 3rd quadrant,)`

`text(or)\ \ x in (pi, (3 pi)/2)(+2k pi, k in ZZ)`

♦ Mean mark 49%.
 


 

`cot(x) = a/y = b\ \ =>\ \ y = a/b`

`text(cosec)(-x) = 1/(sin(-x)) = -1/(sin(x)) =-text(cosec)(x)`

 
`text(In 3rd quadrant:)`

`text(cosec)(x) < 0 \ =>\ \ text(cosec)(-x) > 0`

`:. text(cosec)(-x) = 1/y = b/a`

 
`=>  A`

Trigonometry, SPEC2 2017 VCAA 2 MC

The solutions to  `cos(x) > 1/4 text(cosec)(x)`  for  `x ∈ (0,2pi)\ text(\) {pi}`  are given by

  1. `x ∈ (pi/12,(5pi)/12) ∪ ((5pi)/12,(13pi)/12) ∪ ((17pi)/12,2pi)`
  2. `x ∈ (pi/12,(5pi)/12) ∪ ((13pi)/12,(17pi)/12)`
  3. `x ∈ (pi/12,(5pi)/12) ∪ (pi,(13pi)/12) ∪ ((13pi)/12,2pi)`
  4. `x ∈ (pi/12,(13pi)/12) ∪ ((17pi)/12,2pi)`
  5. `x ∈ (pi/12,(5pi)/12) ∪ (pi,(13pi)/12) ∪ ((17pi)/12,2pi)`
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`E`

Show Worked Solution

`cos(x) > 1/4 text(cosec)(x)`

♦ Mean mark 37%.

`4cos(x) > text(cosec)(x)`
 

`text(Consider:)\ \ 4cos(x) = 1/(sin(x))`

`4cos(x)sin(x)` `= 1`
`2(2sin(x)cos(x))` `= 1`
`2(sin(2x))` `= 1`
`sin(2x)` `= 1/2`
`2x` `= pi/6,(5pi)/6,(13pi)/6,(17pi)/6`
`:. x` `= pi/12,(5pi)/12,(13pi)/12,(17pi)/12\ \ \ (x ∈ (0,2pi)\ text(\) {pi})`

 

`x ∈ (pi/12,(5pi)/12) ∪ (pi,(13pi)/12) ∪ ((17pi)/12,2pi)`

 
`=>E`