3.3dHigher tier1MA1 · Section 3: Sequences, functions and graphs

Trigonometric graphs (sin, cos, tan)

Key learnings

Know the three curve shapes
$y=\sin x$ and $y=\cos x$ wave between $-1$ and $1$ with period $360°$ ( $\cos$ is $\sin$ shifted left by $90°$ ); $y=\tan x$ repeats every $180°$ and has vertical asymptotes at $90°, 270°, \dots$
Use symmetry to get the second solution
In $0°$ – $360°$ most equations have two solutions. For $\sin x = k$ the second is $180° -$ (first); for $\cos x = k$ it is $360° -$ (first).
Key values to remember
$\sin x$ : max $1$ at $90°$ , min $-1$ at $270°$ , zero at $0°,180°,360°$ . $\cos x$ : max $1$ at $0°/360°$ , min $-1$ at $180°$ .

Drawn from real examiner reports.

Calculator in the wrong angle mode

There was evidence of students working in gradians (or radians) rather than degrees, giving nonsense trig values.

Giving only one solution in the range

The calculator returns one angle; students often forgot the curve's symmetry gives a second solution in $0°$ – $360°$ .

Not using the trig functions correctly

Mislabelling or misusing $\sin$ , $\cos$ , $\tan$ — examiners ask students to "know how to use the trigonometric functions correctly".

Sketch the curve and draw the line y = k
Sketch $y = \sin x$ (or $\cos x$ ), draw the horizontal line $y = k$ , and read off every crossing inside the range — this guarantees you find all the solutions.

$y = \sin x$ : starts at 0, max $1$ at $90°$ , back to 0 at $180°$ , min $-1$ at $270°$ , 0 at $360°$ . Period $360°$ .
$y = \cos x$ : starts at $1$ , 0 at $90°$ , $-1$ at $180°$ , 0 at $270°$ , back to $1$ at $360°$ . It is $\sin x$ shifted left $90°$ .
$y = \tan x$ : repeats every $180°$ , with vertical asymptotes at $90°$ and $270°$ .

Foundation

Common calculator error in trig questions?

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