Interpret trigonometric graphs and solve equations
Read sine, cosine and tangent over 0°–360°, sketch their key features and find every solution in the stated interval.
Section 7 · Trigonometry
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the range of y = cos x?
What you will learn
- Recognise sine, cosine and tangent graphs over one turn
- Use symmetry and periodicity to solve equations in 0° ≤ x ≤ 360°
- Include interval endpoints when valid and exclude undefined tangent inputs
Know the key graph landmarks
Sine: (0,0), (90,1), (180,0), (270,−1), (360,0). Cosine: (0,1), (90,0), (180,−1), (270,0), (360,1). Both have range [−1,1] and period 360°.
Tangent has zeros at 0°, 180°, 360° and vertical asymptotes at 90° and 270°. Its period is 180°; it is undefined at the asymptotes. Sketch separate branches rather than joining across an asymptote.
Solve for all angles in the interval
Find the calculator's principal reference angle, then use the graph or quadrant signs to list every solution between 0° and 360° inclusive. Sine is positive in quadrants I and II; cosine in I and IV; tangent in I and III.
Check each answer in the original equation and check endpoints. Use degree mode. An equation may have no solution if its requested sine or cosine value lies outside [−1,1].
Pause and explain
At which angles is tan x undefined in 0°–360°?
Worked example
Solve sin x = 1/2 for 0° ≤ x ≤ 360°.
Show the worked solution
- The reference angle is sin⁻¹(1/2) = 30°.
- Sine is positive in quadrants I and II.
- The solutions are 30° and 180° − 30° = 150°.
Answer x = 30°, 150°.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
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.
Solve cos x = −√2/2 for 0° ≤ x ≤ 360°.
Give me a hint
Use the 45° reference angle and identify where cosine is negative.
Compare my reasoning
- The reference angle is 45°.
- Cosine is negative in quadrants II and III.
- x = 180° − 45° = 135° or 180° + 45° = 225°.
x = 135°, 225°.
Look for these in your work
- Reference angle found
- Correct cosine quadrants used
- Both solutions within interval
Solve tan x = 1 for 0° ≤ x ≤ 360°.
Give me a hint
Tangent's reference angle is 45° and it repeats after 180°.
Compare my reasoning
- tan 45° = 1.
- Tangent is positive in quadrants I and III.
- x = 45° and 45° + 180° = 225°.
x = 45°, 225°.
Look for these in your work
- Reference angle found
- Tangent signs used
- All solutions listed
A rotating signal is modelled by y = 2 cos x. For which angles in one full turn is y = 1? State also the maximum and minimum signal values.
Give me a hint
Divide the equation by 2 and solve a cosine equation over the full stated turn.
Compare my reasoning
- 2 cos x = 1 gives cos x = 1/2.
- Cosine is 1/2 at 60° and 300° in 0°–360°.
- Since −1 ≤ cos x ≤ 1, the signal ranges from −2 to 2.
x = 60°, 300°; maximum 2 and minimum −2.
Look for these in your work
- Equation rearranged
- Both cosine solutions found
- Range scaled correctly
Common mistakes
- Giving only the principal inverse-trig angle
- Using sine's quadrant pattern for cosine or tangent
- Connecting tangent through an asymptote
- Forgetting valid 0° or 360° endpoints
What can you explain now?
How many tangent branches appear between its asymptotes at 90° and 270°?
Compare with the explanation
One continuous increasing branch
Tangent is undefined only at the asymptotes.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why must you use both the reference angle and the graph's quadrant pattern to solve a trigonometric equation over a full turn?
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.