Geometry and trigonometry practice questions (IB Maths AA)

Sample questions

The one free IB Maths AA question on this topic, from the free IB Maths AA SL Paper 1 Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 15 marks · Trigonometry: exact values
   This question is about trigonometry with exact values.
   (a)

   Solve $2\cos x = \sqrt{3}$ for $0 \leq x \leq 2\pi$.

   [3]
   (b)

   In triangle $ABC$, $AB = 6$ cm, $AC = 8$ cm and $B\hat{A}C = \frac{\pi}{3}$. Find the exact area of the triangle.

   [2]
   Show answerFrom the free IB Maths AA SL Paper 1 Practice Exam →

Key terms

The definitions and formulas IB Maths AA expects for geometry and trigonometry, typeset as they appear on the paper.

Radian measure, arc length and sector area

An angle of one radian subtends an arc equal to the radius, so $2\pi$ radians make a full turn. Arc length and sector area take their simplest forms in radians.

$$l = r\theta, \qquad A = \frac{1}{2} r^2 \theta$$

Sine rule and cosine rule

The sine rule links each side to the sine of its opposite angle; the cosine rule generalises Pythagoras. The area of any triangle is half the product of two sides and the sine of the included angle.

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}, \qquad c^2 = a^2 + b^2 - 2ab\cos C, \qquad \text{Area} = \frac{1}{2}ab\sin C$$

Exact trigonometric values

The values at $0$, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$ and $\frac{\pi}{2}$, which Paper 1 expects from memory; they are not in the formula booklet.

$$\sin\frac{\pi}{6} = \frac{1}{2}, \quad \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}, \quad \tan\frac{\pi}{3} = \sqrt{3}, \quad \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$$

Pythagorean and double-angle identities

The identities used to rewrite an equation in a single trigonometric function before solving it. Three forms of $\cos 2\theta$ are in the booklet; choose the one that eliminates the unwanted function.

$$\sin^2\theta + \cos^2\theta = 1, \qquad \sin 2\theta = 2\sin\theta\cos\theta, \qquad \cos 2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$$

Trigonometric equations in an interval

Find the principal solution, then use the symmetry of the unit circle for the others: if $\sin x = k$ has solution $\alpha$, then $\pi - \alpha$ is also a solution, and add or subtract $2\pi$ to stay in the interval given.

Vectors and the scalar product (HL)

The scalar product gives the angle between two vectors and is zero when they are perpendicular. A line is written as a point plus a multiple of a direction vector.

$$\mathbf{a}\cdot\mathbf{b} = |\mathbf{a}|\,|\mathbf{b}|\cos\theta, \qquad \mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$$

Volume and surface area of solids

The formulas for the sphere, cone, pyramid and cylinder, combined in questions about composite solids. The sphere and cone are in the booklet.

$$V_{\text{sphere}} = \frac{4}{3}\pi r^3, \qquad V_{\text{cone}} = \frac{1}{3}\pi r^2 h, \qquad V_{\text{pyramid}} = \frac{1}{3} A h$$

Frequently asked questions

Other IB Maths AA topics

• Number and algebra
• Functions
• Statistics and probability
• Calculus

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