Relate complementary angles without swapping the wrong values
Use sinθ=cos(90°−θ) · Use cosθ=sin(90°−θ) · Explain why complementary angles exchange side roles
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
- Right-triangle ratios
- Acute angle sum
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
- Use sinθ=cos(90°−θ)
- Use cosθ=sin(90°−θ)
- Explain why complementary angles exchange side roles
Exchange the two acute angles
The acute angles of a right triangle sum to 90°. A leg opposite one is adjacent to the other, while the hypotenuse stays fixed. Therefore sinθ=cos(90°−θ) and cosθ=sin(90°−θ). This does not mean sine and cosine of the same angle are always equal.
Use the relation before rounding
For example sin38°=cos52°, so their quotient is exactly 1 because the denominator is non-zero. In a 3–4–5 right triangle with sinθ=3/5, the complementary angle has cosine 3/5, while cosθ=4/5. The unit-circle endpoint identities sin0°=cos90° and cos0°=sin90° agree with the same relations.
Sources for this lesson
Pause and explain
Is sin20° equal to cos70°?
Worked example
Simplify sin38°/cos52° exactly.
Show the worked solution
- 38°+52°=90°, so the numerator and denominator agree.
- They are non-zero, so their quotient is 1.
Answer 1.
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.
Simplify sin38°/cos52° exactly.
Give me a hint
Use the complementary relation first.
Compare my reasoning
- cos52°=sin38°.
- The common value is non-zero, so the quotient is 1.
1.
Look for these in your work
- Use the angle sum
- Check the denominator
Rewrite cos20° and cos70° using sine of an acute angle.
Give me a hint
Subtract each angle from 90°.
Compare my reasoning
- cos20°=sin70°.
- cos70°=sin20°.
sin70°; sin20°.
Look for these in your work
- Keep the angle order
- Use the correct complement
In a 3–4–5 right triangle, sinθ=3/5 and φ=90°−θ. Find cosφ and cosθ.
Give me a hint
The complementary angle swaps the leg roles.
Compare my reasoning
- cosφ=sinθ=3/5.
- For the original θ, adjacent/hypotenuse is 4/5.
cosφ=3/5; cosθ=4/5.
Look for these in your work
- Distinguish the two angles
- Use the right side roles
Common mistakes
- Complementary means the angles sum to 90°, not 180°.
- Do not replace cosine at the complementary angle by cosine at the same angle.
What can you explain now?
Is sin20° equal to cos70°?
Compare with the explanation
Yes; the angles are complementary.
Their opposite and adjacent leg roles exchange in the same right triangle.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Is sin20° equal to cos70°?
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.