Use all six trigonometric functions for any angle
Use the unit circle and quadrant signs to find sine, cosine, tangent and their reciprocals for positive, negative and larger-than-one-turn angles.
Section 10 · Trigonometry
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
On the unit circle, what are the coordinates at angle θ?
What you will learn
- Define all six trigonometric functions for an angle of any magnitude
- Use a reference angle and quadrant to determine exact signs and values
- Recognise when a trigonometric function is undefined
The unit circle extends trigonometry beyond a right triangle
For an angle θ in standard position, its terminal side meets the unit circle at (cos θ, sin θ). Thus sin θ is the y-coordinate and cos θ is the x-coordinate, even when θ is negative or larger than 360° (2π radians).
The other four functions are tan θ = sin θ/cos θ, cot θ = cos θ/sin θ, sec θ = 1/cos θ and cosec θ = 1/sin θ. These definitions also show where a function is undefined: a denominator must not be zero.
Reference angles give exact values; quadrants give signs
Reduce an angle by full turns if helpful, find its acute reference angle, and use the reference-angle values for 30°, 45° or 60° (or π/6, π/4 or π/3). Then apply the sign of x and y in the angle's quadrant.
In Quadrants I and III, sine and cosine have the same sign, so tangent is positive; in II and IV, they have opposite signs, so tangent is negative. The reciprocal functions keep the sign of the function they reciprocate.
For example, 210° has reference angle 30° in Quadrant III. Its sine and cosine are both negative, so its tangent is positive. Since cos 90° = 0, tan 90° and sec 90° are undefined.
Pause and explain
Which two functions are undefined at 90°?
Worked example
Find the exact values of all six trigonometric functions of 210°.
Show the worked solution
- The reference angle is 30° and 210° lies in Quadrant III, where sine and cosine are negative.
- sin 210° = −1/2 and cos 210° = −√3/2.
- tan 210° = (−1/2)/(−√3/2) = 1/√3. Take reciprocals for cot, sec and cosec.
Answer sin = −1/2, cos = −√3/2, tan = 1/√3, cot = √3, sec = −2/√3, cosec = −2.
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.
Find all six exact trigonometric values at 330°.
Give me a hint
Use reference angle 30° and apply the signs in Quadrant IV.
Compare my reasoning
- sin 330° = −1/2 and cos 330° = √3/2.
- tan 330° = −1/√3 and cot 330° = −√3.
- Take reciprocals: sec 330° = 2/√3 and cosec 330° = −2.
sin = −1/2, cos = √3/2, tan = −1/√3, cot = −√3, sec = 2/√3, cosec = −2.
Look for these in your work
- Reference angle is 30°
- Quadrant signs correct
- Reciprocals and undefined cases checked
Find all six exact trigonometric values at 420°.
Give me a hint
Remove one complete turn, then use the 60° values.
Compare my reasoning
- 420° − 360° = 60°.
- At 60°, sin = √3/2, cos = 1/2 and tan = √3.
- Take reciprocals for cot, sec and cosec.
sin = √3/2, cos = 1/2, tan = √3, cot = 1/√3, sec = 2, cosec = 2/√3.
Look for these in your work
- Equivalent angle found
- Six function values correct
- No function is undefined at 60°
A rotating sensor reports an angle of −150°. Find sin, cos and tan, and explain why the negative angle is allowed.
Give me a hint
Measure clockwise from the positive x-axis; the terminal side is coterminal with 210°.
Compare my reasoning
- −150° + 360° = 210°, so the reference angle is 30° in Quadrant III.
- Sine and cosine are negative there: sin(−150°) = −1/2 and cos(−150°) = −√3/2.
- Their ratio is tan(−150°) = 1/√3.
sin = −1/2, cos = −√3/2 and tan = 1/√3. A negative angle records clockwise rotation and is defined on the unit circle.
Look for these in your work
- Negative rotation interpreted
- Coterminal angle and quadrant correct
- Signs and ratio checked
Common mistakes
- Using the reference-angle values without the quadrant signs
- Treating an angle above 360° as invalid instead of periodic
- Taking a reciprocal before checking whether the original value is zero
- Mixing degrees and radians
What can you explain now?
At 300°, which of sin, cos and tan are negative?
Compare with the explanation
sin and tan
300° is in Quadrant IV: sine is negative, cosine positive and tangent negative.
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
How do the unit circle, a reference angle, quadrant signs and reciprocals work together to find all six values?
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.