Circular functions extended (VCE Maths Methods Unit 2)

What this dot point is asking

VCAA wants you to sketch and analyse transformed sine and cosine functions, identifying amplitude, period, phase shift and vertical translation, and to solve trigonometric equations over a specified interval.

Parent functions

$\sin x$ and $\cos x$:

• Domain: all real $x$. Range: $[-1, 1]$.
• Period: $2\pi$. Amplitude: $1$.

Transformed form $y = a\sin(b(x - h)) + k$

Same parameters apply to $y = a\cos(b(x - h)) + k$.

Solving trig equations over a stated interval

1. Isolate the trig function.
2. Find the principal solution.
3. Use symmetry and periodicity to find all solutions in the stated interval.
4. If the equation involves $bx$, list all solutions for $bx$ in the expanded interval and divide by $b$.

Reference angles and quadrant signs

The engine of trig equation solving is the reference angle, the acute angle whose sine, cosine or tangent has the same magnitude as the required value. Find it from the positive value, then place solutions in the correct quadrants using the sign of the original equation: sine is positive in quadrants 1 and 2, cosine in quadrants 1 and 4, and tangent in quadrants 1 and 3. The standard placements are $\pi - \theta$ (quadrant 2), $\pi + \theta$ (quadrant 3) and $2\pi - \theta$ (quadrant 4). Exact values from the special triangles ($\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}$) appear constantly, so commit them to memory for tech-free Exam 1.

Sketching transformed graphs

A reliable sketch shows the centre line $y = k$, the maximum $y = k + |a|$ and minimum $y = k - |a|$, and one full period of width $\dfrac{2\pi}{|b|}$ starting from the phase shift $h$. Mark the five key points across a period (the quarter, half and three-quarter points), since these locate the maxima, minima and crossings of the centre line. Reading $a$, $b$, $h$ and $k$ directly off the equation, then plotting these features, is exactly the method VCAA rewards.

In one sentence

Transformed circular functions $y = a\sin(b(x - h)) + k$ have amplitude $|a|$, period $2\pi/|b|$, phase shift $h$ and centre line $y = k$; solving trig equations over an interval uses the reference angle plus quadrant symmetry, and equations involving $bx$ require finding solutions for $bx$ in the expanded interval before dividing by $b$.

Examples in context

Example 1. Tidal height model. A harbour's water depth (metres) is modelled by $D = 2\sin\!\left(\frac{\pi}{6}t\right) + 5$, where $t$ is hours after midnight. The amplitude is $2$ m about a mean depth of $5$ m, and the period is $\frac{2\pi}{\pi/6} = 12$ hours. High tide ($D = 7$) first occurs when $\sin\!\left(\frac{\pi}{6}t\right) = 1$, i.e. $\frac{\pi}{6}t = \frac{\pi}{2}$, giving $t = 3$ hours (3 am).

Example 2. Ferris wheel height. A rider's height is $h = 10\cos\!\left(\frac{\pi}{30}t\right)\cdot(-1) + 12$, i.e. $h = -10\cos\!\left(\frac{\pi}{30}t\right) + 12$ metres, with $t$ in seconds. The period is $\frac{2\pi}{\pi/30} = 60$ s per rotation, amplitude $10$ m about a centre $12$ m. At $t = 0$, $h = -10 + 12 = 2$ m (boarding height); the maximum $22$ m occurs at $t = 30$ s.

Try this

Q1. State the amplitude, period and range of $y = 3\sin(2x) - 1$. [3 marks]

• Cue. Amplitude $3$; period $\frac{2\pi}{2} = \pi$; range $[-4, 2]$.

Q2. Solve $2\sin(x) = \sqrt{3}$ for $x \in [0, 2\pi]$. [3 marks]

• Cue. $\sin x = \frac{\sqrt 3}{2}$; $x = \frac{\pi}{3}, \frac{2\pi}{3}$.

Q3. A signal is modelled by $y = 4\cos\!\left(\frac{\pi}{2}t\right)$. (a) State its period. (b) Find the first time $t > 0$ when $y = 0$. [2+2 marks]

• Cue. (a) Period $\frac{2\pi}{\pi/2} = 4$. (b) $\cos\!\left(\frac{\pi}{2}t\right) = 0 \Rightarrow \frac{\pi}{2}t = \frac{\pi}{2} \Rightarrow t = 1$.