smc-725-40-Combination of ratios

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

Show Answers Only

`A`

Show Worked Solution

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

♦♦ Mean mark 31%.

`cos(x) = 3/5`

`cos(y) = 12/13`

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

`=> A`

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

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`x = pi/6,\ \ \ -(5 pi)/6`

Show Worked Solution

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

MARKER’S COMMENT: Many students who found the base angle correctly could not solve within the restrictions.

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

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

State all possible values of `tan (theta)`. (1 mark)

--- 7 WORK AREA LINES (style=lined) ---
Hence, find all possible solutions for `(tan (theta)-1) (sin^2 (theta)-3 cos^2 (theta)) = 0`, where `0 <= theta <= pi`. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

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%.

`:. 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`