Read sine and cosine graphs, amplitudes and periods
Read the quarter-turn values on degree graphs · Separate amplitude from period · Distinguish a displayed window from the function domain
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
- Full-turn values
- Function graphs and transformations
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
- Read the quarter-turn values on degree graphs
- Separate amplitude from period
- Distinguish a displayed window from the function domain
A cycle and a vertical range
For y=sinθ in degrees, values at 0°,90°,180°,270°,360° are 0,1,0,−1,0. For y=cosθ they are 1,0,−1,0,1. Both have amplitude 1, range −1≤y≤1 and fundamental period 360°. The curves are smooth between these landmarks, not straight polygonal joins.
Inside and outside factors do different jobs
For y=a sin(bθ) or a cos(bθ), with non-zero a,b and degree input, amplitude is |a| and period 360°/|b|. Thus y=2sin2θ has amplitude 2 and period 180°. A graph shown only over 0° to 360° does not automatically restrict the underlying real-input domain. A zero multiplier gives a constant function, for which there is no least positive period.
Sources for this lesson
Pause and explain
Does multiplying sinθ by 3 change its period from 360°?
Worked example
State amplitude and period of y=2sin2θ with θ in degrees.
Show the worked solution
- The outside factor gives amplitude |2|=2.
- The inside factor gives period 360°/2=180°.
Answer Amplitude 2; period 180°.
Try a topic test
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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.
Use the graph to find sin270° and cos180°.
Give me a hint
Read the minima at the corresponding inputs.
Compare my reasoning
- Sine at 270° is −1.
- Cosine at 180° is −1.
−1; −1.
Look for these in your work
- Read the correct coloured curve
- Use the degree input
For y=2sin2θ, state amplitude, period and all zeros on 0°≤θ≤360°.
Give me a hint
The sine input is zero at multiples of 180°.
Compare my reasoning
- Amplitude=2 and period=180°.
- 2θ=0°,180°,360°,540°,720°, giving θ=0°,90°,180°,270°,360°.
Amplitude 2; period 180°; zeros 0°,90°,180°,270°,360°.
Look for these in your work
- Separate vertical and horizontal factors
- Include both stated endpoints
A graph of cosθ displays only 0°≤θ≤360°. Does the picture alone prove the function is undefined at 450°? Find cos450°.
Give me a hint
Use periodicity rather than the picture edge.
Compare my reasoning
- The underlying cosine function accepts real degree inputs.
- 450°=360°+90°, so cos450°=0.
No; cos450°=0 unless a separate domain restriction is stated.
Look for these in your work
- Distinguish window and domain
- Use the period
Common mistakes
- The horizontal degree scale determines the period units.
- A plotting window is not necessarily a domain restriction.
What can you explain now?
Does multiplying sinθ by 3 change its period from 360°?
Compare with the explanation
No; it changes the amplitude to 3.
The outside multiplier changes heights, while the inside angle factor determines the period.
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Make it stick
Does multiplying sinθ by 3 change its period from 360°?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.