IB Maths AA HL
Topic 3 — Geometry & Trigonometry
Paper 1 & 2
~6 min read
Graphs of Trigonometric Functions
Three graphs to know cold: y = sin x, y = cos x, and y = tan x. Sin and cos are smooth waves that repeat every 360°; tan is a steeper curve with vertical asymptotes that repeats every 180°. Memorise the shapes and key points — you’ll be drawing them constantly from here on.
📘 What you need to know
- y = sin x: passes through origin (0, 0). Period 360° (or 2π). Range [−1, 1]. Amplitude 1.
- y = cos x: passes through (0, 1). Period 360° (or 2π). Range [−1, 1]. Amplitude 1.
- y = tan x: passes through origin. Period 180° (or π). No amplitude. Range = all real numbers.
- Asymptotes of tan: at x = ±90°, ±270°, ±450°, … (or ±π/2, ±3π/2, …). Domain excludes these.
- sin is odd: sin(−x) = −sin x. Rotational symmetry about the origin.
- cos is even: cos(−x) = cos x. Reflective symmetry about the y-axis.
- tan is odd: tan(−x) = −tan x.
- Always check whether the question is in degrees or radians before sketching.
y = sin x and y = cos x
Both have the same shape — a smooth wave between −1 and 1, repeating every 360° (or 2π). They differ only in their starting position: sin starts at 0, cos starts at 1.
| x | 0° | 90° | 180° | 270° | 360° |
|---|
| sin x | 0 | 1 (max) | 0 | −1 (min) | 0 |
| cos x | 1 (max) | 0 | −1 (min) | 0 | 1 (max) |
cos is sin shifted left by 90°: cos x = sin(x + 90°). Same shape, different starting point. This is why they have identical periods, ranges, and amplitudes.
y = tan x
Different shape and different period. The graph rises from −∞ to +∞ across each interval of length 180°, with vertical asymptotes between intervals.
Tan — key features
Period: 180° (π) Range: ℝ Asymptotes: x = 90° + 180°k
| x | 0° | 45° | 90° | 135° | 180° |
|---|
| tan x | 0 | 1 | undefined | −1 | 0 |
Odd vs even — the symmetries
sin and tan — ODD
sin(−x) = −sin x
tan(−x) = −tan x
rotational symmetry about origin
cos — EVEN
cos(−x) = cos x
reflective symmetry about the y-axis
🧭 Recipe — sketching a trig graph
- Check units: degrees or radians? Set up axes accordingly.
- Mark the x-axis in multiples of 90° (or π/2), covering the whole interval.
- Mark the y-axis: [−1, 1] for sin/cos; full range for tan.
- Plot key points at every multiple of 90° (or π/2). For tan, mark the asymptotes.
- Join smoothly for sin/cos; for tan, draw the curve approaching each asymptote.
Worked examples
WE 1Read values from y = sin x
For the graph of y = sin x, find (a) the value at x = 270°, (b) the value at x = 540°, and (c) all values of x in 0° ≤ x ≤ 720° where sin x = 0.
(a) Read directly
sin 270° = −1 (minimum point)
(a) y = −1
(b) Use periodicity (period 360°)
540° − 360° = 180°
sin 540° = sin 180° = 0
(b) y = 0
(c) sin x = 0 at every multiple of 180°
(c) x = 0°, 180°, 360°, 540°, 720°
WE 2Use the symmetries of y = sin x
Given that sin 70° ≈ 0.9397, use the properties of the sine graph to find (a) sin(−70°), (b) sin 110°, and (c) sin 250°.
(a) Sin is odd
sin(−70°) = −sin 70° ≈ −0.9397
(b) 110° = 180° − 70° (Q2 symmetry)
sin 110° = sin 70° ≈ 0.9397
(c) 250° = 180° + 70° (Q3)
sin 250° = −sin 70° ≈ −0.9397
(a) ≈ −0.940; (b) ≈ 0.940; (c) ≈ −0.940
no calculator needed beyond the given value — symmetries do all the work
WE 3Properties of y = tan x
Consider the graph of y = tan x in radians. Find (a) the period, (b) the x-coordinates of all asymptotes in 0 ≤ x ≤ 3π, and (c) the value of tan(5π/4).
(a) Tan repeats every π
(a) period = π
(b) Asymptotes at π/2 + kπ
in [0, 3π]: π/2, 3π/2, 5π/2
(b) x = π/2, 3π/2, 5π/2
(c) Use period π
tan(5π/4) = tan(5π/4 − π) = tan(π/4) = 1
(c) tan(5π/4) = 1
WE 4Number of solutions in an interval
Without solving, find the number of solutions of sin x = 0.4 in the interval 0° ≤ x ≤ 1080°.
Step 1: Number of full periods in the interval
1080° ÷ 360° = 3 periods
Step 2: 0 < 0.4 < 1 → two solutions per period
total = 3 × 2 = 6
6 solutions
draw the horizontal line y = 0.4 across three full waves to see this visually
WE 5Sketch y = sin x and y = cos x together
On the same axes, sketch the graphs of y = sin x and y = cos x for 0 ≤ x ≤ 2π. Mark all x-intercepts and all maximum and minimum points.
y = sin x — key points
x-intercepts: (0, 0), (π, 0), (2π, 0)
maximum: (π/2, 1)
minimum: (3π/2, −1)
y = cos x — key points
x-intercepts: (π/2, 0), (3π/2, 0)
maxima: (0, 1) and (2π, 1)
minimum: (π, −1)
Where they meet
(π/4, √2/2) and (5π/4, −√2/2)
two waves; cos peaks where sin crosses zero, and vice versa
WE 6Identify a function from its properties
A function f(x) has period 180°, passes through the origin, and has vertical asymptotes at every odd multiple of 90°. Identify f(x) from among y = sin x, y = cos x, and y = tan x.
Step 1: Eliminate by period
sin and cos have period 360° → ruled out
Step 2: Check tan
tan x has period 180° ✓
tan 0 = 0 → passes through origin ✓
asymptotes at ±90°, ±270°, … ✓
f(x) = tan x
period is the fastest discriminator — sin/cos = 360°; tan = 180°
💡 Top tips
- Sketch the three base graphs at the start of every trig question. Free reference for the rest of the paper.
- Always label key points: x-intercepts, maxima, minima, asymptotes — and the units (degrees or radians).
- Use periodicity: any angle can be reduced into the first period by adding/subtracting 360° (or π for tan).
- Sin and cos are translations of each other: cos x = sin(x + 90°). Useful for proofs and transformations.
- Number of solutions: divide the interval length by the period, then multiply by solutions-per-period (2 for sin/cos, 1 for tan).
⚠ Common mistakes
- Mixing degrees and radians on the axes. Pick one and stick with it.
- Forgetting tan asymptotes. The graph never touches x = 90°, 270°, etc — it shoots off to ±∞.
- Confusing sin and cos starting points. sin starts at 0; cos starts at 1.
- Assuming tan has period 360°. It’s 180° — half the period of sin/cos.
- Drawing tan as a wave. It’s a steep, repeating curve through the origin, not a wave between −1 and 1.
Next note: Solving Equations Using Trigonometric Graphs. With the graphs in your head, you can find every solution to sin x = k, cos x = k, or tan x = k in a given interval — just draw the horizontal line and read off where it cuts the curve.
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