Read amplitude and period from trigonometric graphs
Describe a sin(bx), a cos(bx) and tan x in degrees · Separate amplitude from period · Recognise tangent's excluded angles
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Sine, cosine and tangent values
- Function graphs in degrees
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Describe a sin(bx), a cos(bx) and tan x in degrees
- Separate amplitude from period
- Recognise tangent's excluded angles
Two independent scales
For non-zero a and b, a sin(bx) and a cos(bx) have amplitude |a| and period 360°/|b| when x is in degrees. The sign of a reflects the graph vertically. A greater |b| shortens the cycle; it does not change its height.
For y = 3 sin(2x), zeros occur when 2x is a multiple of 180°. One complete cycle fits 0° to 180°: key points (0°, 0), (45°, 3), (90°, 0), (135°, −3), (180°, 0). Cosine begins at its amplitude for a positive a.
Tangent has a different shape
tan x has period 180°, zeros at integer multiples of 180° and vertical asymptotes at 90° + 180°k. It has no amplitude: outputs have no finite maximum or minimum. Never connect the branches across an excluded angle.
If a or b is zero, the sine/cosine rule may produce a constant function; the usual non-zero formula's least positive period then does not apply. State the angle unit before reading a period.
Sources for this lesson
Pause and explain
What is the period of sin(2x) in degrees?
Worked example
Find the amplitude and period of y = −2 cos(3x), with x in degrees.
Show the worked solution
- Amplitude is |−2| = 2; the negative sign reflects the cosine graph.
- Period is 360°/3 = 120°. At x = 0, the output is −2.
Answer Amplitude 2, period 120°, starting value −2.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For y = 4 sin(2x), give amplitude, period and the first positive maximum input.
Give me a hint
A sine maximum needs 2x = 90°.
Compare my reasoning
- Amplitude is 4; period is 180°.
- The first positive maximum occurs at x = 45°.
Amplitude 4; period 180°; first maximum at 45°.
Look for these in your work
- Separate input and output scales
- Use degrees
For y = −3 cos(x/2), give its period and outputs at x = 0°, 180° and 360°.
Give me a hint
The cosine input is half the stated x.
Compare my reasoning
- Period is 360°/(1/2) = 720°.
- Cosine inputs are 0°, 90°, 180°, giving outputs −3, 0, 3.
Period 720°; outputs −3, 0, 3.
Look for these in your work
- Retain the reflection
- Divide by the input scale
A plotted tan x curve connects (80°, tan 80°) straight through 90° to (100°, tan 100°). Explain the error and give its zeros in 0° ≤ x ≤ 360°.
Give me a hint
Check where the denominator cos x is zero.
Compare my reasoning
- tan x is undefined at 90° and 270°, so those branches are separated by vertical asymptotes.
- Zeros are 0°, 180° and 360°.
Do not connect through 90° or 270°; zeros 0°, 180°, 360°; period 180° and no finite amplitude.
Look for these in your work
- Mark excluded angles
- Include allowed interval endpoints
Common mistakes
- Tangent does not have amplitude 1.
- A sine coefficient of 2 changes height; an input coefficient of 2 changes period.
What can you explain now?
What is the period of sin(2x) in degrees?
Compare with the explanation
180°.
The input angle completes 360° when x increases by 180°.
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Make it stick
What is the period of sin(2x) in degrees?
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