Graphs of sin, cos & tan

For $y = a\sin(bx) + c$ (and the $\cos$ twin):

• Amplitude $= |a|$, height from the centre line to a peak
• Period $= \dfrac{360^\circ}{b}$ (or $\frac{2\pi}{b}$ in radians), one full cycle’s width
• Centre line: $y = c$, the curve oscillates between $c - |a|$ and $c + |a|$

These read directly off the equation, and “state the amplitude and period of $y = 3\sin 2x - 1$” is a write-down command: amplitude $3$, period $180^\circ$, no working expected.

The three base shapes

$\sin x$: starts at $0$, peaks at $90^\circ$, wave through $(0,0)$, $(180^\circ, 0)$, $(360^\circ, 0)$. $\cos x$: the same wave shifted, starts at the peak $(0, 1)$. $\tan x$: period $180^\circ$, no amplitude, through the origin, asymptotes at $90^\circ + 180^\circ n$, and a $\tan$ sketch without its asymptotes drawn dashed is missing its defining feature.

Sketching transformed graphs

Sketch $y = 2\cos 3x + 1$ for $0^\circ \le x \le 180^\circ$. Amplitude $2$, period $120^\circ$, centre line $y = 1$ $\to$ oscillates between $-1$ and $3$. Starts at the maximum $(0, 3)$; completes $1\frac{1}{2}$ cycles by $180^\circ$. Label: max/min values, the centre line, and the $x$-axis crossings.

Marks attach to labelled features: maxima/minima with coordinates, intercepts, period structure visible (the right number of cycles in the window). Count cycles before drawing, $b = 3$ in a $180^\circ$ window means exactly $1\frac{1}{2}$ periods, and the wrong cycle count is the most common sketch error.

Negative $a$ reflects in the centre line; $y = |\sin x|$ folds per the modulus rules.

Graphs as solution counters

The exam’s favourite payoff: “state the number of solutions of $2\cos 3x + 1 = 2$ for $0^\circ \le x \le 180^\circ$”, draw $y = 2$ across the sketch, count intersections. The graph answers in seconds what algebra answers in minutes (the graphical-solving principle); when the question says using your sketch, the count is the expected method. Conversely, knowing the period tells you how many solutions a trig equation should produce, a built-in completeness check.

Common mistakes

• Period computed as $360^\circ \times b$ instead of $\div\, b$
• Amplitude read as the max value when $c \ne 0$ shifts the wave
• Wrong cycle count in the window
• $\tan$ drawn without asymptotes (or with an amplitude)
• $\sin$/$\cos$ starting points swapped

Full topic context: Trigonometry notes.