Recall exact trigonometric values
Use the exact sine and cosine values at 0°, 30°, 45°, 60° and 90°, and tangent values through 60°.
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 exact value of sin 30°?
What you will learn
- Recall exact sine and cosine values at the five specified angles
- Recall exact tangent values at 0°, 30°, 45° and 60°
- Use symmetry and exact surd forms without premature decimals
Build the familiar exact-value table
For x = 0°, 30°, 45°, 60°, 90°, sin x = 0, 1/2, √2/2, √3/2, 1; cos x = 1, √3/2, √2/2, 1/2, 0.
For x = 0°, 30°, 45°, 60°, tan x = 0, 1/√3 (equivalently √3/3), 1, √3. The syllabus does not list tan 90° as a required exact value; it is undefined because cos 90° = 0.
Connect ratios to triangles and symmetry
The 30°–60°–90° triangle has side ratio 1:√3:2; the 45°–45°–90° triangle has ratio 1:1:√2. These produce the exact forms above.
Keep exact surds and fractions in symbolic answers. If a calculator decimal is requested, convert only at the end.
Pause and explain
Which is an equivalent exact form of tan 30°?
Worked example
Find the exact value of sin 60° + cos 30° + tan 45°.
Show the worked solution
- sin 60° = √3/2 and cos 30° = √3/2.
- tan 45° = 1.
- Add the exact values: √3/2 + √3/2 + 1 = √3 + 1.
Answer 1 + √3.
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.
Evaluate exactly: 2 sin 30° + cos 0°.
Give me a hint
Use the exact table before multiplying.
Compare my reasoning
- sin 30° = 1/2, so 2 sin 30° = 1.
- cos 0° = 1.
- The sum is 2.
2.
Look for these in your work
- Exact values recalled
- Coefficient applied
- Sum simplified
Find the exact value of cos 45° × tan 60°.
Give me a hint
Use √2/2 for cos 45° and √3 for tan 60°.
Compare my reasoning
- cos 45° × tan 60° = (√2/2)(√3).
- Multiply the surds: √6/2.
- This is already an exact simplified form.
√6/2.
Look for these in your work
- Both table entries correct
- Surds multiplied
- Exact form retained
A unit circle has a point at 30°. State its exact coordinates (cos 30°, sin 30°), then give the exact tangent of the angle.
Give me a hint
On the unit circle x = cos θ and y = sin θ; tangent is y/x.
Compare my reasoning
- cos 30° = √3/2 and sin 30° = 1/2.
- The point is (√3/2, 1/2).
- tan 30° = (1/2)/(√3/2) = 1/√3 = √3/3.
(√3/2, 1/2); tan 30° = √3/3.
Look for these in your work
- Coordinates ordered cosine then sine
- Unit circle interpreted
- Tangent ratio simplified
Common mistakes
- Swapping sine and cosine rows
- Writing 1/√3 as √3 without rationalising or explaining
- Giving a decimal when an exact form is required
- Assigning a finite value to tan 90°
What can you explain now?
What is the exact value of cos 60°?
Compare with the explanation
1/2
The 60° entry is half in the cosine row.
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Make it stick
How can the special triangle side ratios help you reconstruct exact values instead of memorising disconnected numbers?
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.