Consider the function `g: R -> R, \ g(x) = 2sin(2x).`
- State the range of `g`. (1 mark)
- State the period of `g`. (1 mark)
- Solve `2 sin(2x) = sqrt3` for `x ∈ R`. (3 marks)
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Consider the function `g: R -> R, \ g(x) = 2sin(2x).`
a. `text(S)text(ince) -1<sin(2x)<1,`
`text(Range)\ g(x) = [–2,2]`
b. `text(Period) = (2pi)/n = (2pi)/2 = pi`
c. | `2sin(2x)` | `=sqrt3` |
`sin(2x)` | `=sqrt3/2` | |
`2x` | `=pi/3, (2pi)/3, pi/3 + 2pi, (2pi)/3 + 2pi, …` | |
`x` | `=pi/6, pi/3, pi/6+pi, pi/3+pi, …` |
`:.\ text(General solution)`
`= pi/6 + npi, pi/3 + npi\ \ \ (n in ZZ)`
Solve the equation
`qquad sin (2x + pi/3) = 1/2\ \ text(for)\ \ x in [0, pi].` (2 marks)
`x = pi/4, (11 pi)/12`
`sin (2x + pi/3) = 1/2`
`=>\ text(Base angle is)\ \ pi/6`
`(2x + pi/3)` | `= pi/6, (5pi)/6, (13pi)/6, (17pi)/6, …` |
`2x` | `= – pi/6, pi/2, (11pi)/6, (15pi)/6, …` |
`x` | `= – pi/12, pi/4, (11pi)/12, (15pi)/12, …` |
`:. x = pi/4, (11 pi)/12,\ \ x in [0, pi]`
The sum of the solutions of `sin(2x) = (sqrt3)/2` over the interval `[−pi,d]` is `−pi`.
The value of `d` could be
`C`
`sin(2x)` | `=sqrt3/2` |
`2x` | `=pi/3, (2pi)/3, ..` |
`x` | `=pi/6, pi/3, ..` |
`text(Expand and test possible solutions to solve:)`
`x=-(5pi)/6, -(2pi)/3, pi/6, pi/3, ..`
`-(5pi)/6+ -(2pi)/3+ pi/6+ pi/3=-pi`
`=> C`
The general solution to the equation `sin (2x) = -1` is
`A`
`2x` | `= 2n pi – pi/2,\ \ n in Z` |
`x` | `= n pi – pi/4,\ \ n in Z` |
`=> A`
Let `f: R -> R`, `f(x) = sin((2pix)/3)`.
a. `sin((2pix)/3) = -sqrt3/2`
`=>\ text(Base angle)\ = pi/3`
`(2 pi x)/3` | `=(4pi)/3, (5pi)/3, (10pi)/3, …` |
`:.x` | `=2 or 5/2, \ \ \ text(for)\ x ∈ [0,3]` |
b. `g(x) = 3sin[(2pi)/3 (x – 1)] + 2`
`text(Maximum occurs when:)`
`sin[(2pi)/3 (x – 1)]` | `= 1` |
`(2pi)/3 (x – 1)` | `= pi/2` |
`x-1` | `= pi/2 xx 3/(2pi)` |
`:. x` | `=7/4` |
Solve the equation `sin (x/2) = -1/2` for `x in [2 pi, 4 pi].` (2 marks)
`x = (7 pi)/3, (11 pi)/3`
`x/2` | `=pi/6 + pi, 2pi – pi/6, 2pi + (pi/6 +pi), …` |
`=(7pi)/6, (11pi)/6, (19pi)/6, …` | |
`:. x` | `=(7pi)/3, (11pi)/3, (19pi)/3, …` |
`text(Given)\ \ x in [2 pi, 4 pi],`
`:. x = (7 pi)/3, (11 pi)/3`
On any given day, the depth of water in a river is modelled by the function
`h(t) = 14 + 8sin((pit)/12),\ \ 0 <= t <= 24`
where `h` is the depth of water, in metres, and `t` is the time, in hours, after 6 am.
a. `h_(text(min))\ text(occurs when)\ \ sin((pit)/12)=-1`
`:. h_(text(min))` | `= 14 – 8` |
`= 6\ text(m)` |
b. | `14 + 8sin(pi/12t)` | `= 10` |
`sin(pi/12t)` | `= – 1/2` |
`text(Solve in general:)`
`pi/12t` | `=(7pi)/6 + 2pi n\ \ \ \ text(or)\ \ \ ` | `pi/12t` | `= (11t)/6 + 2pi n,` |
`t` | `= 14 + 24n` | `t` | `=22 + 24n` |
`text(Substitute integer values for)\ n,`
`:. t = 14quadtext(or)quad22,\ \ \ (t ∈ [0,24])`