Trigonometric Graphs and Transformations
Read and sketch sine, cosine and tangent graphs with amplitude, vertical shift, frequency and phase change.
How to study National 5 Mathematics
Secure one operational skill at a time, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and a checked conclusion.
Core concepts
Concept 1
Basic sine and cosine graphs have period 360°; tangent has period 180°.
Exam cue: Identify the midline, maximum, minimum and period before finding parameters.
Concept 2
In y = a sin(bx) + c or y = a cos(bx) + c, |a| controls amplitude, b controls period and c gives vertical translation.
Exam cue: Use period = 360°/|b| for sine or cosine and 180°/|b| for tangent.
Concept 3
A phase angle moves the graph horizontally and must be interpreted with the equation’s sign and scale.
Exam cue: Check phase direction by testing a key point rather than relying only on a memorised sign rule.
Risk pitfalls and guardrails
Calling vertical translation the amplitude.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Using 360° as the tangent period.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Reading x- and y-coordinates interchangeably.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Memory anchors
Sine period
360° for the basic graph.
Cosine period
360° for the basic graph.
Tangent period
180° for the basic graph.
Amplitude
Half the distance between maximum and minimum for sine or cosine.
Vertical translation
The midline value.
Multiple angle
Sine/cosine period 360°/|b|; tangent period 180°/|b|.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Find sin 30°.
Find cos 60°.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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