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.
By Timothy Gan6 December 2021Singapore O-Level syllabus
Free revision pack
Use the notes online, then download the worksheet to practise without distractions.
- 5 concept sections
- 5 worked questions
- 10 video lessons
5
Concept sections
5
Practice questions
5
Concept videos
5
Solution videos
Start here
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.
Video lesson
Study Guide - Sketching of Sine Curve
1
Concept
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.
Video lesson
Sketching cosine curves
Video lesson
Sketching tangent curves
2
Concept
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.
If $a<0$, the graph is reflected in the $x$-axis, while its amplitude remains $|a|$. Tangent graphs do not have amplitude because they are unbounded.
Video lesson
Study Guide - Sketching Trigonometric Curves with Negative Amplitude
3
Concept
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 $720^\circ$.
4
Concept
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.
Video lesson
Study Guide - Sketching of Translated Curves
5
Concept
A Reliable Sketching Process
Use this process for most A Math trigonometric curve questions:
- Identify the basic graph: sine, cosine or tangent.
- Find the amplitude and period.
- Mark the midline and key x-values.
- Plot the maximum, minimum, intercepts or asymptotes.
- Draw a smooth curve through the key points.
Always label axes, intercepts, turning points and asymptotes when they are relevant.
Guided practice
Practice Questions with Video Solutions
Attempt each question before opening the solution. Then compare your method with the worked steps and video explanation.
Question 1Video solution
Sketching a Sine Curve
Question
Sketch the graph $y=5\sin3x$ for $0^\circ\le x\le360^\circ$, stating the amplitude and period.
Video lesson
Study Guide - Trigonometric Curves Question 1
Step-by-step solution
- 1Compare the function with $y=a\sin bx$: the amplitude is $|a|=5$.
- 2The period is $\frac{360^\circ}{3}=120^\circ$.
- 3The interval contains $\frac{360^\circ}{120^\circ}=3$ complete cycles.
- 4Mark quarter-period points every $30^\circ$, with maximum $5$, minimum $-5$, and intercepts on the midline $y=0$, then draw three smooth sine cycles.
Final answer
Amplitude $5$, period $120^\circ$, with 3 complete cycles
Question 2Video solution
Sketching a Cosine Curve
Question
Sketch the graph of $y=5\cos3x$ for $0^\circ\le x\le240^\circ$, stating the amplitude and period.
Video lesson
Study Guide - Trigonometric Curves Question 2
Step-by-step solution
- 1The amplitude is $|5|=5$.
- 2The period is $\frac{360^\circ}{3}=120^\circ$.
- 3The interval contains $\frac{240^\circ}{120^\circ}=2$ complete cycles.
- 4Start at the maximum point $(0^\circ,5)$, mark quarter-period points every $30^\circ$, and repeat the cosine pattern for two cycles.
Final answer
Amplitude $5$, period $120^\circ$, with 2 complete cycles
Question 3Video solution
Sketching a Tangent Curve
Question
Sketch the graph of $y=\tan(3x)$ for $0^\circ\le x\le210^\circ$. State the period of the function.
Video lesson
Study Guide - Trigonometric Curves Question 3
Step-by-step solution
- 1For $y=\tan bx$, the period is $\frac{180^\circ}{|b|}=\frac{180^\circ}{3}=60^\circ$.
- 2The interval contains $\frac{210^\circ}{60^\circ}=3.5$ cycles.
- 3Vertical asymptotes occur when $3x=90^\circ+180^\circ k$, so $x=30^\circ+60^\circ k$.
- 4Draw increasing tangent branches between the asymptotes and mark intercepts where $x=60^\circ k$.
Final answer
Period $60^\circ$, with 3.5 cycles over the stated interval
Question 4Video solution
Vertical Translation of a Sine Curve
Question
The equation of a curve is $y=2\sin3x-1$ for $0\le x\le2\pi$.
(i) Write down the maximum and minimum values of $y$.
(ii) State the amplitude, period and range.
(iii) Hence, sketch the graph.
(i) Write down the maximum and minimum values of $y$.
(ii) State the amplitude, period and range.
(iii) Hence, sketch the graph.
Video lesson
Study Guide - Trigonometric Curves Question 4
Step-by-step solution
- 1The midline is $y=-1$ and the amplitude is $2$.
- 2Maximum value $=2(1)-1=1$; minimum value $=2(-1)-1=-3$.
- 3The period is $\frac{2\pi}{3}$ and the range is $-3\le y\le1$.
- 4The interval contains $\frac{2\pi}{2\pi/3}=3$ complete cycles. Sketch three sine cycles about the midline $y=-1$.
Final answer
Maximum $1$, minimum $-3$, amplitude $2$, period $\frac{2\pi}{3}$, range $-3\le y\le1$
Question 5Video solution
Negative Amplitude and Reflection
Question
Sketch the graph of $y=-5\cos2x+3$ for $0\le x\le3\pi$.
Video lesson
Study Guide - Trigonometric Curves Question 5
Step-by-step solution
- 1The amplitude is $|-5|=5$ and the negative coefficient reflects the cosine graph in the $x$-axis.
- 2The period is $\frac{2\pi}{2}=\pi$, so the interval contains 3 complete cycles.
- 3The vertical translation gives the midline $y=3$.
- 4The maximum is $3+5=8$ and the minimum is $3-5=-2$. At $x=0$, the graph starts at the minimum value $-2$.
Final answer
Amplitude $5$, period $\pi$, midline $y=3$, maximum $8$, minimum $-2$
Revision summary
Key Formulas to Remember
Sine and Cosine Amplitude
For y = a sin x or y = a cos x, amplitude = |a|
Sine and Cosine Period
For y = sin bx or y = cos bx, period = 360 degrees / |b|
Tangent Period
For y = tan bx, period = 180 degrees / |b|
General Sine Transformation
y = a sin b(x - c) + d
General Cosine Transformation
y = a cos b(x - c) + d
Identities and Results to Memorise
- y = sin x has period 360 degrees
- y = cos x has period 360 degrees
- y = tan x has period 180 degrees
- amplitude = |a| for sine and cosine
- midline = y = d
- period = 360 degrees / |b| for sine and cosine
Make Trigonometry Less Guesswork
Our A Math tuition classes teach students to sketch graphs from structure, not memory alone, so trigonometry becomes more predictable in exams.
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