Circular (trig) functions and graphs: VCE Math Methods Unit 3

What this dot point is asking

VCAA wants you to graph and analyse the three primary circular functions $\sin$, $\cos$ and $\tan$ in radians, recognise their key features under transformation, and quote exact values at standard unit-circle angles without a calculator. This is core Paper 1 content.

Sine and cosine

$f(x) = \sin(x)$ and $f(x) = \cos(x)$ are the two foundational trig functions.

Sine.

• Domain: $\mathbb{R}$. Range: $[-1, 1]$.
• Period: $2\pi$. Amplitude: $1$.
• x-intercepts at $x = n\pi$ for $n \in \mathbb{Z}$.
• Starts at $\sin(0) = 0$, rises to maximum $1$ at $x = \pi/2$, returns to $0$ at $x = \pi$, minimum $-1$ at $x = 3\pi/2$, back to $0$ at $x = 2\pi$.

Cosine.

• Domain: $\mathbb{R}$. Range: $[-1, 1]$.
• Period: $2\pi$. Amplitude: $1$.
• x-intercepts at $x = \pi/2 + n\pi$.
• Starts at $\cos(0) = 1$, drops to $0$ at $x = \pi/2$, minimum $-1$ at $x = \pi$, back to $0$ at $x = 3\pi/2$, max $1$ at $x = 2\pi$.

Cosine is sine shifted left by $\pi/2$: $\cos(x) = \sin(x + \pi/2)$.

Tangent

$f(x) = \tan(x) = \frac{\sin(x)}{\cos(x)}$.

• Domain: $\mathbb{R} \setminus \{\pi/2 + n\pi : n \in \mathbb{Z}\}$ (excluded where $\cos(x) = 0$).
• Range: $\mathbb{R}$.
• Period: $\pi$ (not $2\pi$).
• Vertical asymptotes at $x = \pi/2 + n\pi$.
• x-intercepts at $x = n\pi$.

Within one period, the tangent graph rises steeply from negative infinity through the origin to positive infinity.

Exact values at standard angles

These are Paper 1 essential. Memorise the unit-circle table:

For angles beyond $[0, \pi/2]$, use the ASTC quadrant rule:

• Quadrant 1 ($0$ to $\pi/2$): all positive.
• Quadrant 2 ($\pi/2$ to $\pi$): sine positive only.
• Quadrant 3 ($\pi$ to $3\pi/2$): tangent positive only.
• Quadrant 4 ($3\pi/2$ to $2\pi$): cosine positive only.

The reference angle (acute angle to the x-axis) gives the magnitude; the quadrant gives the sign.

Transformed circular functions

The standard form is $f(x) = a \sin(b(x - h)) + k$ (and similarly for $\cos$).

• Amplitude $= |a|$. Vertical dilation by factor $|a|$; reflection in the x-axis if $a < 0$.
• Period $= \frac{2\pi}{|b|}$. Horizontal dilation by factor $\frac{1}{|b|}$.
• Horizontal translation $= h$ (right if $h > 0$).
• Midline (vertical translation) $y = k$.
• Range: $[k - |a|, k + |a|]$.

For tan, the period under $\tan(b(x - h))$ is $\frac{\pi}{|b|}$ instead of $2\pi/|b|$.

The Pythagorean identity

Useful corollaries:

• $\tan^2(x) + 1 = \sec^2(x)$ where $\sec(x) = \frac{1}{\cos(x)}$ (used in $\frac{d}{dx}(\tan x) = \sec^2 x$).
• $\cos^2(x) = 1 - \sin^2(x)$ and $\sin^2(x) = 1 - \cos^2(x)$ for simplifying.

Examples in context

Example 1. Modelling temperature over a day. A city's temperature is modelled by $T = 18 - 6\cos\!\left(\frac{\pi}{12}t\right)$ degrees Celsius, where $t$ is hours after midnight. The amplitude is $6$ about a mean of $18$, with period $\frac{2\pi}{\pi/12} = 24$ hours. The minimum ($T = 12$) occurs at $t = 0$ (midnight, where $\cos = 1$) and the maximum ($T = 24$) at $t = 12$ (noon).

Example 2. Exact value with the ASTC rule. To evaluate $\tan\!\left(\frac{4\pi}{3}\right)$: the angle is in the third quadrant ($\pi + \frac{\pi}{3}$) with reference angle $\frac{\pi}{3}$. Tangent is positive in quadrant 3, and $\tan\!\left(\frac{\pi}{3}\right) = \sqrt 3$, so $\tan\!\left(\frac{4\pi}{3}\right) = \sqrt 3$.

Try this

Q1. State the amplitude, period and range of $y = 4\sin(3x) + 2$. [3 marks]

• Cue. Amplitude $4$; period $\frac{2\pi}{3}$; range $[-2, 6]$.

Q2. Find the exact value of $\cos\!\left(\frac{5\pi}{4}\right)$. [2 marks]

• Cue. Quadrant 3, reference $\frac{\pi}{4}$, cosine negative: $-\frac{\sqrt 2}{2}$.

Q3. For $y = 2\sin\!\left(2\!\left(x - \frac{\pi}{4}\right)\right)$, state (a) the period and (b) the phase shift. [2+1 marks]

• Cue. (a) $\frac{2\pi}{2} = \pi$. (b) $\frac{\pi}{4}$ to the right.