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Algebra, MET2 2021 VCAA 16 MC

Let  `cos (x) = 3/5`  and  `sin^2(y) = 25/169`, where  `x ∈ [{3pi}/{2} , 2 pi]`  and  `y ∈ [{3pi}/{2} , 2 pi]`.

The value of  `sin(x) + cos(y)`  is 

  1. `8/65`
  2. `– 112/65`
  3. `112/65`
  4. `–8/65`
  5. `64/65`
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`A`

Show Worked Solution

`text{Both angles are in 4th quadrant (given)}`

♦♦ Mean mark 31%.

`cos(x) = 3/5`
 

`sin(x)` `= – 4/5\ \ text{(4th quadrant)}`
`sin^2(y)` `= 25/169`
`sin(y)` `= – 5/13\ \ text{(4th quadrant)}`

 

`cos(y) = 12/13`
 

`:. \ sin(x) + cos(y)` `= – 4/5 + 12/13`
  `= 8/65`

`=> A`

Functions, MET1 2010 VCAA 4b

Solve the equation  `sqrt 3 sin (x) = cos (x)`  for  `x in [– pi, pi]`.  (2 marks)

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`x = pi/6,\ \ \ – (5 pi)/6`

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`text(Divide both sides by)\ \ cos(x) :`

MARKER’S COMMENT: Many students who found the base angle correctly could not solve within the restrictions.
`sqrt 3 sin x` `=cos x`
`sqrt 3 tan x` `= 1`
`tan x` `= 1/sqrt 3`
`=>\ text(Base angle)\ = pi/6`

 

`:. x = pi/6\ \ text(or)\ \ – (5 pi)/6,\ \ \ x in [– pi, pi].`

Algebra, MET1 2017 VCAA 6

Let  `(tan(theta) - 1) (sin (theta) - sqrt 3 cos (theta)) (sin (theta) + sqrt 3 cos(theta)) = 0`.

  1. State all possible values of  `tan (theta)`.  (1 mark)
  2. Hence, find all possible solutions for  `(tan (theta) - 1) (sin^2 (theta) - 3 cos^2 (theta)) = 0`, where  `0 <= theta <= pi`.  (2 marks)
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  1. `tan theta = 1 or tan theta = +- sqrt 3`
  2. `theta = pi/4, pi/3 or (2 pi)/3`
Show Worked Solution

a.  `(tan(theta) – 1) (sin (theta) – sqrt 3 cos (theta)) (sin (theta) + sqrt 3 cos(theta)) = 0`

`=> tan theta = 1`

♦ Mean mark 42%.
`=>sin theta – sqrt 3 cos theta` `=0`
`sin theta` `=sqrt3 cos theta`
`tan theta` `=sqrt3`

 

`=>sin theta + sqrt 3 cos theta` `=0`
`sin theta` `=-sqrt3 cos theta`
`tan theta` `=-sqrt3`

 

`:. tan theta = 1 or tan theta = +- sqrt 3`

 

b.  `(tan (theta) – 1) (sin^2 (theta) – 3 cos^2 (theta)) = 0`

`text(Using part a:)`

♦ Mean mark 42%.

`(tan theta – 1) (sin theta – sqrt 3 cos theta) (sin theta + sqrt 3 cos theta) = 0`

`=> tan theta` `= 1` `qquad or qquad` `tan theta` `= +- sqrt 3`
`theta` `= pi/4`   `theta` `= pi/3, (2 pi)/3\ \ text(for)\ \ theta ∈ [0, pi]`

 

`:. theta = pi/4, pi/3 or (2 pi)/3`