Trigonometric graphs over one full turn
Sketch sine, cosine and tangent on 0° to 360°, marking zeros, turning points and asymptotes.
Extended only.
Before you begin
- Read coordinates and axis scales.
- Know special-angle values.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is sin 90°?
What you will learn
- Recognise and sketch the three basic degree graphs.
- Distinguish continuous sine/cosine waves from tangent's separate branches.
Sine and cosine repeat after 360°
Sine passes through (0°,0), (90°,1), (180°,0), (270°,−1) and (360°,0). Cosine passes through (0°,1), (90°,0), (180°,−1), (270°,0) and (360°,1). Both have range −1 to 1 and period 360°. Mark these key points, then connect with a smooth curve, not straight line segments.
The horizontal scale measures degrees, while the vertical scale measures a dimensionless ratio. Cosine is not sine reflected in the horizontal axis: their starting values differ. A horizontal line y = k can reveal how many solutions a sine or cosine equation has on the requested interval.
Tangent has breaks, not high turning points
Tangent has zeros at 0°, 180° and 360° and vertical asymptotes at 90° and 270°. Between the asymptotes it rises, approaching arbitrarily large positive or negative values; its period is 180°. It is undefined at the asymptotes, so do not draw a joining line through either break.
Tangent is positive in the first and third quadrants and negative in the second and fourth. Sine is positive in the upper half-turn; cosine is positive in the right half-turn. These sign regions and the basic graph shapes help locate all equation solutions rather than accepting only a calculator's principal answer.
Pause and explain
Where does y = tan x have vertical asymptotes on 0° to 360°?
Worked example
List the zeros and undefined angles of y = tan x for 0° ≤ x ≤ 360°.
Show the worked solution
- A zero requires sin x = 0 while cos x is nonzero, giving 0°, 180° and 360°.
- Undefined points occur when cos x = 0, at 90° and 270°.
- Keep separate graph branches at those two asymptotes.
Answer Zeros 0°, 180°, 360°; undefined at 90°, 270°.
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.
Sketch y = sin x over 0° to 360°. Mark its five quarter-turn points.
Give me a hint
Start at zero and pass through positive then negative extrema.
Compare my reasoning
- Plot (0,0), (90,1), (180,0), (270,−1), (360,0).
- Use labelled degree and ratio axes.
- Draw a smooth wave through the points without exceeding ±1.
Sine wave through the five listed points.
Look for these in your work
- Marked the correct extrema and zeros.
- Used a smooth curve on paper.
Sketch y = cos x over the same interval and state its minimum point.
Give me a hint
Cosine starts at one rather than zero.
Compare my reasoning
- Plot (0,1), (90,0), (180,−1), (270,0), (360,1).
- Join smoothly within the range −1 to 1.
- The minimum point is (180°,−1).
Cosine wave; minimum (180°, −1).
Look for these in your work
- Used the correct starting point.
- Located the negative turning point.
A student joins tangent's branches with a vertical stroke at 90°. Explain and correct the error.
Give me a hint
The asymptote marks values at which tangent is undefined.
Compare my reasoning
- No tangent value exists at ninety degrees.
- Draw the separate branches approaching the asymptote, without a connecting stroke.
- Also mark the break at 270° and keep zeros at 0°, 180°, 360°.
Separate branches at 90° and 270°; no connecting vertical stroke.
Look for these in your work
- Distinguished an asymptote from part of the graph.
- Included both breaks in the interval.
Common mistakes
- Drawing sine as straight segments.
- Starting cosine at zero.
- Joining tangent across an asymptote.
What can you explain now?
What is the range of y = sin x?
Compare with the explanation
−1 ≤ y ≤ 1.
The sine ratio cannot exceed one in magnitude and attains both endpoint values during a full turn.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, sketch the three graphs from key values, identifying which has discontinuities.
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources