Practice IB Mathematics Analysis and Approaches (AA) Topic Geometry & Trigonometry with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for Geometry & Trigonometry and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

SLPaper 1

Prove the identity
$\frac{1-\tan^2\theta}{1+\tan^2\theta}=\cos(2\theta),$ for all values of $θ$ for which both sides are defined.

Start from the left-hand side and express everything in terms of $sin θ$ and $cos θ$ .

[3]

Hence, or otherwise, show that $sin (2 θ) = \frac{2 t a n θ}{1 + t a n ^{2} θ} .$

[3]

If $tan θ = \frac{3}{4}$ , find the exact values of $sin (2 θ)$ and $cos (2 θ)$ .

[4]

Question 2

HLPaper 2

Consider the function $f (x) = ln (2 x - 1) - sin (x)$ , where $x > \frac{1}{2}$ .

Sketch the graph of $y = f (x)$ for $0.5 < x \leq 4$ , showing the intercepts and the behavior at $x = \frac{1}{2}$ .

[3]

Solve the inequality $ln (2 x - 1) < sin (x)$ .

[4]

Question 3

SLPaper 2

A hiker runs 620 m on a bearing of $22 6^{\circ}$ to a checkpoint, and then 360 m on a bearing of $13 6^{\circ}$ to the finish.

Find the direct distance from the starting point to the finishing point.

[3]

Find the bearing of the finishing point from the starting point.

[3]

Question 4

HLPaper 1

Solve the equation $3 csc^{2} x = 4$ in the interval $[0, 2 π]$ .

[6]

Question 5

SLPaper 1

Solve the equation $tan (3 π x - \frac{π}{4}) = tan π x$ in the interval $[0, 2]$ .

[8]

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