IB · Mathematics: Analysis & Approaches SL & HL · Geometry & trigonometry

Trig identities & exact values: IB Mathematics: Analysis & Approaches questions with answers

Free Trig identities & exact values questions for IB Mathematics: Analysis & Approaches (SL & HL), part of Geometry & trigonometry. Each question has the answer and the marking points an examiner would look for.

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Trig identities & exact values practice questions

Question 1Structured · 4 marks

Show that 1 − cos 2xsin 2x = tan x.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
shown
use of cos 2x = 1 − 2sin2x M1
use of sin 2x = 2sin x cos x M1
2sin2x2sin x cos x A1
= sin xcos x = tan x AG A1
4
Question 2Structured · 7 marks

Let cos θ = −817, where π2 < θ < π.

(a)

Find the exact value of sin θ.

[2]
(b)

Find the exact value of sin 2θ.

[2]
(c)

Find the exact value of tan 2θ.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)1517
use of sin2θ + cos2θ = 1 (M1)
sin θ = 1517 A1
2
(b)−240289
2(1517)(−817) (M1)
−240289 A1
2
(c)240161
cos 2θ = −161289 (M1)(A1)
tan 2θ = 240161 A1
3
Question 3Structured · 3 marks

Show that (sin x + cos x)2 = 1 + sin 2x.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
shown
expanding: sin2x + 2sin x cos x + cos2x M1
sin2x + cos2x = 1 A1
2sin x cos x = sin 2x AG A1
3
Question 4Structured · 7 marks

Let cos θ = −513, where π2 < θ < π.

(a)

Find the exact value of sin θ.

[2]
(b)

Find the exact value of sin 2θ.

[2]
(c)

Find the exact value of tan 2θ.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)1213
use of sin2θ + cos2θ = 1 (M1)
sin θ = 1213 A1
2
(b)−120169
2(1213)(−513) (M1)
−120169 A1
2
(c)120119
cos 2θ = −119169 (M1)(A1)
tan 2θ = 120119 A1
3
Question 5Structured · 3 marks

Show that cos4x − sin4x = cos 2x.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
shown
difference of squares: (cos2x − sin2x)(cos2x + sin2x) M1
cos2x + sin2x = 1 A1
cos2x − sin2x = cos 2x AG A1
3
Question 6Structured · 7 marks

Let sin θ = 35, where π2 < θ < π.

(a)

Find the exact value of cos θ.

[2]
(b)

Find the exact value of sin 2θ.

[2]
(c)

Find the exact value of tan 2θ.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)−45
use of sin2θ + cos2θ = 1 (M1)
cos θ = −45 (negative in the second quadrant) A1
2
(b)−2425
2(35)(−45) (M1)
−2425 A1
2
(c)−247
cos 2θ = 725 (M1)(A1)
tan 2θ = −247 A1
3

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in