Graphs of circular functions $f(x) = \sin(x)$, $f(x) = \cos(x)$ and $f(x) = \tan(x)$, their key features (period, amplitude, asymptotes), exact values at standard angles, and graphs of the form $f(x) = a\sin(b(x - h)) + k$

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:

• IMATH_90 where $\sec(x) = \frac{1}{\cos(x)}$ (used in $\frac{d}{dx}(\tan x) = \sec^2 x$).
• IMATH_93 and $\sin^2(x) = 1 - \cos^2(x)$ for simplifying.

Worked example

State the amplitude, period and range of $f(x) = -3 \sin(2x - \pi/3) + 1$, and find the y-intercept.

Rewrite in standard form: $f(x) = -3 \sin\!\left(2\!\left(x - \frac{\pi}{6}\right)\right) + 1$.

Amplitude: $|-3| = 3$. Period: $\frac{2\pi}{2} = \pi$. Midline: $y = 1$. Range: $[1 - 3, 1 + 3] = [-2, 4]$.

y-intercept: $f(0) = -3 \sin(-\pi/3) + 1 = -3 \cdot (-\sqrt{3}/2) + 1 = \frac{3\sqrt{3}}{2} + 1$.

Common Paper 1 traps

Period mismatch for $\tan$. $\tan(b x)$ has period $\frac{\pi}{b}$, not $\frac{2\pi}{b}$. Easy 1-mark loss.

Wrong sign from the quadrant. $\sin(7\pi/6)$ is negative (quadrant 3), not positive. Always sketch the unit circle or use ASTC.

Degrees instead of radians. All VCE Math Methods trig is in radians. Writing $\sin(30)$ instead of $\sin(\pi/6)$ will be marked wrong.

Forgetting the horizontal dilation reciprocal. $b = 2$ compresses horizontally by factor 2, halving the period.

Including too many or too few solutions. When solving $\sin(2x) = 1/2$ for $x \in [0, 2\pi]$, you have $2x \in [0, 4\pi]$, so four solutions for $2x$, hence four solutions for $x$.

In one sentence

The circular functions sine, cosine and tangent are the trigonometric backbone of VCE Math Methods, with sine and cosine bounded between $-1$ and $1$ with period $2\pi$, tangent unbounded with period $\pi$ and vertical asymptotes, and all three transforming under the standard form $a\sin(b(x-h)) + k$ to amplitude $|a|$, period $\frac{2\pi}{|b|}$, and midline $y = k$.