Syllabus map

AA SL Geometry and Trigonometry syllabus

The checklist below outlines all Standard Level syllabus subtopics for Topic 3. Each lesson guide features interactive theory, 10 tiered exercises, formula summaries, and exam-style tests.

  1. 3.13D Coordinate geometry: distance formula, midpoint, volume and surface area
  2. 3.2Right-triangle trigonometry, sine rule, cosine rule, area formula $\frac{1}{2}ab\sin C$
  3. 3.3Applications of trigonometry: bearings, angles of elevation/depression, 3D solids
  4. 3.4Circle geometry: radian measure, arc length $s = r\theta$, sector area $A = \frac{1}{2}r^2\theta$
  5. 3.5Unit circle exact values and the ambiguous case of the sine rule (SSA)
  6. 3.6Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$, quotient identity, double-angle formulas
  7. 3.7Trigonometric functions: $f(x) = a\sin(b(x-c))+d$, amplitude, period, sinusoidal modeling
  8. 3.8Solving trigonometric equations analytically and graphically over finite domains

Study by topic

Geometry and Trigonometry lessons

Work through focused interactive lesson guides. Each contains complete study notes, semantic diagrams, 10 tiered problems with collapsible solutions, formula booklets, and mini tests.

3.1–3.3Geometry and Triangle Applications3D coordinate distance, right-triangle ratios, non-right triangles (sine and cosine rules), area, bearings, and angles in 3D solids.Open lesson 3.4Circles and RadiansRadian conversion, arc length $s = r\theta$, sector area $A = \frac{1}{2}r^2\theta$, and circular segment calculations.Open lesson 3.5Exact Values and Ambiguous Sine RuleUnit circle definition, exact values for standard angles, ASTC quadrants, and the ambiguous SSA triangle case.Open lesson 3.6Trigonometric IdentitiesPythagorean identity $\sin^2\theta + \cos^2\theta = 1$, $\tan\theta = \frac{\sin\theta}{\cos\theta}$, and double-angle formulas for sine and cosine.Open lesson 3.7Trigonometric Functions & TransformationsGraphs of sine, cosine, and tangent; transformations $a\sin(b(x-c))+d$; period $\frac{2\pi}{b}$, amplitude $|a|$, and periodic real-world models.Open lesson 3.8Solving Trigonometric EquationsAnalytical and graphical solutions using reference angles, unit circle symmetries, quadratic trig equations, and multi-argument angles.Open lesson

Learn the ideas

Key ideas and methods

Diagrams, triangles and circle measure

  • A clear diagram distinguishes known lengths and angles from unknowns. In right triangles, apply SOH CAH TOA or Pythagoras. In non-right triangles, apply the Sine Rule $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ or the Cosine Rule $a^2 = b^2 + c^2 - 2bc\cos A$.
  • Angle bearings are measured clockwise from True North ($000^\circ$ to $360^\circ$) using 3 digits.
  • Radian measure is defined by $\theta = \frac{s}{r}$. In radians, arc length is $s = r\theta$, sector area is $A = \frac{1}{2}r^2\theta$, and segment area is $A = \frac{1}{2}r^2(\theta - \sin\theta)$.

Exact trigonometric values and identities

  • Recall exact values: $\sin\frac{\pi}{6} = \frac{1}{2}$, $\cos\frac{\pi}{3} = \frac{1}{2}$, $\sin\frac{\pi}{4} = \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$, $\tan\frac{\pi}{4} = 1$. Quadrant signs follow ASTC (All, Sine, Tangent, Cosine).
  • The fundamental Pythagorean identity is $\sin^2\theta + \cos^2\theta = 1$. The double-angle formulas are $\sin 2\theta = 2\sin\theta\cos\theta$ and $\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$.
  • The ambiguous sine rule case (SSA) arises when $a < b$ and $a > b\sin A$, yielding two valid angles: acute $B_1$ and obtuse $B_2 = 180^\circ - B_1$ (provided $A + B_2 < 180^\circ$).

Trigonometric graphs and equations

  • For $y = a\sin(b(x - c)) + d$, amplitude is $|a|$, period is $\frac{2\pi}{|b|}$ (or $\frac{\pi}{|b|}$ for tangent), principal axis is $y = d$, and horizontal phase shift is $c$.
  • When solving trigonometric equations over finite domains $[0, 2\pi]$, identify reference angles, locate active quadrants, and adjust intervals when dealing with multi-arguments (e.g. $2x$ requires checking $[0, 4\pi]$).

Follow the reasoning

Worked examples

These original examples show common reasoning steps and standard IB notation.

Worked example 1

Find an arc length and sector area

Question. A circle has radius $r = 6\text{ cm}$ and a sector angle of $\theta = \frac{5\pi}{6}\text{ rad}$. Calculate the arc length and the area of the sector in exact form.

  1. $s = r\theta = 6 \cdot \frac{5\pi}{6} = 5\pi\text{ cm}$.
  2. $A = \frac{1}{2}r^2\theta = \frac{1}{2}(36)\left(\frac{5\pi}{6}\right) = 18 \cdot \frac{5\pi}{6} = 15\pi\text{ cm}^2$.

Answer: Arc length $5\pi\text{ cm}$; area $15\pi\text{ cm}^2$.

Worked example 2

Check the ambiguous case of the sine rule

Question. In $\triangle ABC$, $a = 8\text{ cm}$, $A = 30^\circ$, and $b = 12\text{ cm}$. Find the possible values of angle $B$.

  1. Apply the Sine Rule: $\frac{\sin B}{b} = \frac{\sin A}{a} \implies \sin B = \frac{12\sin 30^\circ}{8} = \frac{12(0.5)}{8} = 0.75$.
  2. Acute solution: $B_1 = \arcsin(0.75) \approx 48.6^\circ$.
  3. Obtuse candidate: $B_2 = 180^\circ - 48.6^\circ = 131.4^\circ$.
  4. Verification: $A + B_2 = 30^\circ + 131.4^\circ = 161.4^\circ < 180^\circ$, so both triangles are geometrically valid.

Answer: $B \approx 48.6^\circ$ or $B \approx 131.4^\circ$.

Worked example 3

Solve a trigonometric equation on an interval

Question. Solve $2\sin x = \sqrt{3}$ for $0 \le x < 2\pi$.

  1. Isolate the trig ratio: $\sin x = \frac{\sqrt{3}}{2}$.
  2. The reference angle is $x_{\text{ref}} = \frac{\pi}{3}$.
  3. Since $\sin x > 0$, solutions lie in Quadrants I and II:
    • Quadrant I: $x = \frac{\pi}{3}$
    • Quadrant II: $x = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$

Answer: $x = \frac{\pi}{3}$ or $x = \frac{2\pi}{3}$.

Try it yourself

Practice with worked answers

  1. Question 1. A sector of a circle of radius $4\text{ cm}$ subtends an angle of $1.2\text{ rad}$. Calculate its perimeter and area.

    Show worked answer

    Arc length $s = r\theta = 4(1.2) = 4.8\text{ cm}$. Total perimeter $P = 2r + s = 8 + 4.8 = 12.8\text{ cm}$. Area $A = \frac{1}{2}(4^2)(1.2) = 9.6\text{ cm}^2$.

  2. Question 2. Solve $\cos x = \frac{1}{2}$ for $0 \le x < 2\pi$.

    Show worked answer

    Reference angle is $\frac{\pi}{3}$. Cosine is positive in Quadrants I and IV: $x = \frac{\pi}{3}$ and $x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}$.

  3. Question 3. In right-angled triangle $PQR$, the hypotenuse is $13\text{ cm}$ and one leg is $5\text{ cm}$. Find the exact length of the other leg and the acute angle $\theta$ opposite the $5\text{ cm}$ leg to 1 decimal place.

    Show worked answer

    Pythagorean theorem: $\text{leg} = \sqrt{13^2 - 5^2} = \sqrt{144} = 12\text{ cm}$. Angle $\sin\theta = \frac{5}{13} \implies \theta = \arcsin\left(\frac{5}{13}\right) \approx 22.6^\circ$.

Avoid these slips

Common mistakes

  • Using degree values directly in $s = r\theta$ or $A = \frac{1}{2}r^2\theta$ without converting to radians first.
  • Assuming the Ambiguous Case always produces two triangles without verifying $A + B_2 < 180^\circ$.
  • Forgetting that $\cos 2x = 0$ over $[0, 2\pi]$ doubles the number of solutions because $2x \in [0, 4\pi]$.
  • Using the Sine Rule when the Cosine Rule is required (e.g., given three sides SSS or two sides and included angle SAS).

Before you move on

Revision checklist

  • Can I calculate distances, midpoints, and angles in 3D cuboids and pyramids?
  • Can I fluently convert between degrees and radians and calculate arc lengths, sector areas, and segment areas?
  • Can I apply the Sine Rule (including the ambiguous case) and Cosine Rule accurately?
  • Can I manipulate trigonometric identities ($\sin^2\theta+\cos^2\theta=1$, double angles) to solve quadratic trigonometric equations?
  • Can I extract amplitude, period, and phase shifts from sinusoidal modeling functions?

About this guide

This is an independent SJMaths study resource, not an official IB publication. It follows the current AA guide first assessed in 2021 (final assessment November 2028).

Check the IB Mathematics programme page and the official AA subject guide for the authoritative syllabus.