IB DP Maths AA : AHL 3.11 Relationships between trigonometric functions HL Paper 1

(a) The given function \( f(x) = \sin(qx) \), where \( q > 0 \), implies that the function is a sinusoidal function with a period dependent on the value of \( q \). To find an expression for \( m \) in terms of \( q \), we must consider the properties of the sine function and its periodic nature.
For the sine function, the period \( T \) is the length of one complete cycle, which is typically \( 2\pi \) for \( \sin(x) \). However, for \( \sin(qx) \), the period is scaled by a factor of \( \frac{1}{q} \), making the period:
\(
T = \frac{2\pi}{q}
\)
This period can be divided into four equal intervals to find the \( x \)-intercepts of the function, since a complete sine wave crosses the \( x \)-axis at equally spaced intervals within its period.
Given that the \( x \)-intercepts occur at \( x = 0, 2m, 4m \), and the period is \( 4m \), we can equate \( 4m \) to \( T \) to find an expression for \( m \). Therefore, we set:
\(
4m = \frac{2\pi}{q}
\)
and solve for \( m \):
Dividing both sides by 4 gives:
\(
m = \frac{2\pi}{4q}
\)
Simplifying the fraction, we obtain:
\(
m = \frac{\pi}{2q}
\)

(b)To sketch the graph of \( g(x) = 3\sin\left(\frac{2\pi x}{3}\right) \) for \( 0 \leq x \leq 6m \), we first note that the function is a transformation of the parent function \( \sin(x) \).
The coefficient 3 indicates a vertical stretch by a factor of 3, and the term \( \left(\frac{2\pi x}{3}\right) \) indicates a horizontal stretch by a factor of \( \frac{3}{2} \) and a period change to \( 3m \). The period of the function \( g(x) \) is the length of one complete cycle of the sine curve, which can be calculated using the formula:
\(
\text{Period} = \frac{2\pi}{|B|}
\)
where \( B \) is the coefficient of \( x \) in the argument of the sine function. Therefore, the period of \( g(x) \) is \( 3m \).
The amplitude of \( g(x) \) is the absolute value of the coefficient in front of the sine function, which is 3. This means the graph will reach a maximum of 3 units above the midline and a minimum of 3 units below the midline.
The graph of \( g(x) \) will intersect the \( x \)-axis at intervals of:
\(
\frac{3m}{2}
\)
which is half the period. Hence, the \( x \)-intercepts are at:
\(
x = 0, 1.5m, 3m, 4.5m, \text{ and } 6m
\)
The maximum points will occur at:
\(
x = 0.75m \text{ and } x = 3.75m, \text{ with } y = 3
\)
and the minimum points will occur at:
\(
x = 2.25m \text{ and } x = 5.25m, \text{ with } y = -3
\)
To sketch, we start at the origin \( (0,0) \), rise to the first maximum at \( (0.75m, 3) \), cross the \( x \)-axis at \( (1.5m, 0) \), reach the first minimum at \( (2.25m, -3) \), cross the \( x \)-axis again at \( (3m, 0) \), rise to the second maximum at \( (3.75m, 3) \), and continue this pattern until reaching \( (6m, 0) \).
We make sure to plot these key points and draw a smooth, continuous curve that alternates symmetrically above and below the \( x \)-axis, creating the characteristic ‘S’ shape of the sine function.