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O Level (Sec 3 & 4)Additional Mathematics

Sketching of Trigonometric Curves

Learn how to sketch sine, cosine and tangent curves for O-Level A Math. This guide covers amplitude, period, phase shift, vertical translation and graph transformations with a free worksheet and worked solutions.

Timothy Gan

14 May 2026

Download Free Worksheet (PDF)

Reading and Sketching Trigonometric Graphs

Trigonometric graphs show how sine, cosine and tangent values change as the angle changes. For O-Level Additional Mathematics, you should be able to sketch a curve quickly, identify its key features, and explain how transformations affect the graph.

The three most important features are amplitude, period and displacement. Once you can read these from the equation, sketching becomes much more systematic.

Basic Sine, Cosine and Tangent Graphs

The basic sine curve $y=\sin x$ starts at 0, reaches 1, returns to 0, reaches -1 and returns to 0 over $360^\circ$.

The basic cosine curve $y=\cos x$ starts at 1, falls to 0, reaches -1, returns to 0 and reaches 1 again over $360^\circ$.

The tangent curve $y=\tan x$ has a period of $180^\circ$ and vertical asymptotes at $90^\circ$, $270^\circ$ and every $180^\circ$ after that.

Amplitude

For graphs in the form $y=a\sin x$ or $y=a\cos x$, the amplitude is $|a|$.

Amplitude measures the maximum distance from the midline. For example, $y=3\sin x$ has amplitude 3, so its maximum value is 3 and its minimum value is -3.

Tangent graphs do not have amplitude because they are unbounded.

Period

For $y=\sin bx$ or $y=\cos bx$, the period is:

$\frac{360^\circ}{|b|}$

For $y=\tan bx$, the period is:

$\frac{180^\circ}{|b|}$

For example, $y=\sin 2x$ completes one cycle in $180^\circ$, while $y=\cos \frac{x}{2}$ completes one cycle in $720^\circ$.

Phase Shift and Vertical Translation

A graph in the form $y=a\sin b(x-c)+d$ has:

amplitude $|a|$
period $\frac{360^\circ}{|b|}$ for sine and cosine
horizontal shift $c$
vertical shift $d$

The line $y=d$ becomes the new midline. Sketch this midline first, then place the maximum and minimum values around it.

A Reliable Sketching Process

Use this process for most A Math trigonometric curve questions:

1. Identify the basic graph: sine, cosine or tangent. 2. Find the amplitude and period. 3. Mark the midline and key x-values. 4. Plot the maximum, minimum, intercepts or asymptotes. 5. Draw a smooth curve through the key points.

Always label axes, intercepts, turning points and asymptotes when they are relevant.

Practice Questions with Worked Solutions

Work through each question carefully, then compare your method with the step-by-step solution.

Question 1

Finding Amplitude and Period

Question

State the amplitude and period of $y=4\sin 3x$.

Step-by-Step Solution

1Compare $y=4\sin 3x$ with $y=a\sin bx$.
2The amplitude is $|a|=4$.
3The period is $\frac{360^\circ}{|b|}=\frac{360^\circ}{3}=120^\circ$.

Answer:Amplitude = 4, period = $120^\circ$

Question 2

Reading a Vertical Translation

Question

For $y=2\cos x+3$, state the midline, maximum value and minimum value.

Step-by-Step Solution

1The vertical translation is +3, so the midline is $y=3$.
2The amplitude is 2.
3Maximum value = $3+2=5$.
4Minimum value = $3-2=1$.

Answer:Midline $y=3$, maximum value 5, minimum value 1

Question 3

Sketching a Cosine Curve

Question

Describe the key points needed to sketch $y=\cos 2x$ for $0^\circ \le x \le 360^\circ$.

Step-by-Step Solution

1The amplitude is 1.
2The period is $\frac{360^\circ}{2}=180^\circ$.
3The graph completes two full cycles from $0^\circ$ to $360^\circ$.
4Key points are $(0^\circ,1)$, $(45^\circ,0)$, $(90^\circ,-1)$, $(135^\circ,0)$, $(180^\circ,1)$, then the pattern repeats.

Answer:Two cosine cycles with period $180^\circ$ and amplitude 1

Question 4

Tangent Asymptotes

Question

Find the vertical asymptotes of $y=\tan 2x$ for $0^\circ \le x \le 360^\circ$.

Step-by-Step Solution

1The tangent function has asymptotes when its angle is $90^\circ+180^\circ k$.
2For $y=\tan 2x$, set $2x=90^\circ+180^\circ k$.
3Then $x=45^\circ+90^\circ k$.
4Within $0^\circ \le x \le 360^\circ$, the asymptotes are $45^\circ$, $135^\circ$, $225^\circ$ and $315^\circ$.

Answer:$x=45^\circ,135^\circ,225^\circ,315^\circ$