IGCSE · Mathematics 0580 · Trigonometry

Exact trigonometric values: IGCSE Mathematics questions with answers

Free Exact trigonometric values questions for IGCSE Mathematics (0580), Extended tier (Supplement content), part of Trigonometry. Each question has the answer and the marking points an examiner would look for.

Generate a full Exact Trigonometric Values worksheet →All Trigonometry questions

Exact trigonometric values practice questions

Question 1Structured · 3 marksA* challenge

Without using a calculator, find the exact value of
sin2 60° − cos2 60°.
Show all your working.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
12
two correct exact values used B1
correct combination M1
12 A1
3
Question 2Structured · 10 marksA* challenge

From a point A on level ground, the angle of elevation of the top of a vertical tower is 30°. From a point B, 20 m closer to the tower on the same straight line, the angle of elevation is 45°.

(a)

Find the exact height of the tower, with a rational denominator.

[5]
(b)

A different tower of unknown height h stands at F. From P the elevation of its top is 30°, and from Q it is 45°. P and Q are 20 m apart on level ground and angle PFQ = 60°. Find h exactly.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)h = 20 ÷ (√3 − 1) = 10(√3 + 1)
BF = h (45°) B1
tan 30° = hh + 20 = 1√3 M1
h√3 = h + 20 A1
h = 20√3 − 1 M1
rationalised to 10(√3 + 1) A1
5
(b)h = 20√(4 − √3) ≈ 13.3 m
FP = h√3, FQ = h B1
cosine rule in triangle PFQ M1
400 = 3h2 + h2 − 2 × h√3 × h × 12 A1
h2(4 − √3) = 400 M1
h = 20√(4 − √3) A1
5
Question 3Structured · 3 marks
(a)

Write down the exact value of sin 45°.

[1]
(b)

Write down the exact value of tan 60°.

[1]
(c)

Write down the exact value of tan 30°.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)√22
√22 oe B1
1
(b)√3
√3 oe B1
1
(c)√33
√33 oe B1
1
Question 4Structured · 3 marksA* challenge

Without using a calculator, find the exact value of
2 sin 30° + cos2 30°.
Show all your working.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
74
two correct exact values used B1
correct combination M1
74 A1
3
Question 5Structured · 3 marks
(a)

Write down the exact value of cos 30°.

[1]
(b)

Write down the exact value of tan 45°.

[1]
(c)

Write down the exact value of tan 30°.

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)√32
√32 oe B1
1
(b)1
1 oe B1
1
(c)√33
√33 oe B1
1
Question 6Structured · 3 marksA* challenge

Without using a calculator, find the exact value of
(sin 45° + cos 45°)2.
Show all your working.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
2
two correct exact values used B1
correct combination M1
2 A1
3

More Trigonometry questions

Pythagoras’ theoremRight-angled trianglesSine & cosine rules

Want a mentor, not just a paper?

Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.

Meet Riyaz →

Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in