IB Maths AA SLTopic 3 — Geometry & TrigPaper 1 & 2~9 min read
Graphs of Trigonometric Functions
The graphs of y = sin x, y = cos x, and y = tan x are the foundation of every trig topic that follows. Learn their shapes, their key features, and how to sketch them — these come up everywhere from solving trig equations to modelling tides and Ferris wheels.
📘 What you need to know
Sin and cos graphs are waves — same shape, just shifted. Both have amplitude 1, period 360° (2π), and range −1 ≤ y ≤ 1.
sin x passes through the origin (0, 0); cos x passes through (0, 1).
Tan graph has asymptotes at ±90°, ±270° etc. (or ±π/2, ±3π/2 in radians). Period 180° (π). Range = all real numbers.
All three are periodic — they repeat forever.
Sin is an odd function: sin(−x) = −sin x. Cos is an even function: cos(−x) = cos x.
The sine and cosine graphs
The graphs of sin and cos are essentially the same wave — just shifted along the x-axis. Both oscillate smoothly between −1 and 1, repeating every 360° (or 2π radians).
y = sin x and y = cos x on the same axes
Both graphs oscillate between −1 and 1, repeating every 360°.
Cosine = sine shifted left by 90°. If you take the sine curve and slide it 90° to the left, you land exactly on the cosine curve. They’re the same wave, different starting positions.
The tangent graph
The tan graph looks completely different — it’s not a wave. Instead, it shoots up to infinity, “breaks” at vertical lines called asymptotes, then reappears from negative infinity. The asymptotes happen at ±90°, ±270°, etc.
y = tan x
The tan graph has vertical asymptotes at every ±90°, repeating every 180°.
🤔 Why does tan have asymptotes?
Because tan θ = sin θ / cos θ. At 90°, cos 90° = 0 — so tan 90° involves dividing by zero, which is undefined. As x approaches 90° from below, cos x gets tiny while sin x stays close to 1, so the ratio shoots off to +∞. From the other side, it shoots off to −∞. That’s an asymptote.
Comparing the three graphs
Property
y = sin x
y = cos x
y = tan x
Period
360° (2π)
360° (2π)
180° (π)
Amplitude
1
1
none
Range
−1 ≤ y ≤ 1
−1 ≤ y ≤ 1
all real ℝ
y-intercept
0
1
0
Asymptotes?
none
none
at ±90°, ±270°…
Symmetry
odd
even
odd
Key points to know on each graph
y = sin x
starts at 0, peaks at 1 (90°), back to 0 (180°), troughs at −1 (270°), back to 0 (360°).
y = cos x
starts at 1, drops to 0 (90°), troughs at −1 (180°), back to 0 (270°), returns to 1 (360°).
y = tan x
starts at 0, climbs to ∞ (asymptote at 90°), reappears from −∞, back to 0 at 180°, repeats.
Symmetry — odd vs even
The sin and tan graphs have rotational symmetry about the origin (they’re “odd” functions). The cos graph has reflective symmetry in the y-axis (it’s “even”). Knowing this lets you turn negative angles into positive ones quickly.
Odd functions (sin, tan)
Rotational symmetry about the origin.
sin(−x) = −sin x
tan(−x) = −tan x
Even function (cos)
Reflective symmetry in the y-axis.
cos(−x) = cos x
e.g. cos(−60°) = cos 60° = ½
How to sketch a trig graph (5-step method)
1
CHECK MODE
Are angles in degrees or radians? Check the domain.
2
x-AXIS
Mark multiples of 90° (or π/2). Cover the whole interval.
3
y-AXIS
For sin/cos: −1 to 1. For tan: full y-axis with asymptotes.
4
KEY VALUES
Mark sin/cos = 0, ±1 at each multiple of 90°.
5
DRAW
Smoothly join the points. For tan, draw inside each “branch” between asymptotes.
For sin and cos, a good shortcut: mark the maximum, minimum, and zero crossings first — every 90° those alternate. Then connect them with a smooth wave shape. No need to plot dozens of points.
Worked examples
WE 1
Read values off the sin graph
Without using a calculator, find: (a) sin 0° (b) sin 90° (c) sin 180° (d) sin 270° (e) sin 360°.
Step 1: Recall the wave shape
Sin starts at 0, peaks at 1, back to 0, troughs at −1, back to 0.
Step 2: Read off the values(a) sin 0° = 0(b) sin 90° = 1 (the peak)(c) sin 180° = 0(d) sin 270° = −1 (the trough)(e) sin 360° = 00, 1, 0, −1, 0memorise the pattern at the four “quarter points” of one cycle
WE 2
Use periodicity to find values
Given that sin 30° = 0.5, find: (a) sin 390° (b) sin 750° (c) cos 360° (d) tan 540°.
(a) Sin repeats every 360°sin 390° = sin(390 − 360) = sin 30° = 0.5(b) Subtract 360° twicesin 750° = sin(750 − 720) = sin 30° = 0.5(c) Cos repeats every 360°cos 360° = cos 0° = 1(d) Tan repeats every 180°tan 540° = tan(540 − 360) = tan 180° = 0(a) 0.5 (b) 0.5 (c) 1 (d) 0always reduce the angle to within one period first
WE 3
Use odd/even symmetry
Given sin 20° ≈ 0.342 and cos 20° ≈ 0.940, find sin(−20°) and cos(−20°).
Step 1: Sin is odd, cos is evensin(−x) = −sin x | cos(−x) = cos xStep 2: Applysin(−20°) = −sin 20° = −0.342cos(−20°) = cos 20° = 0.940sin(−20°) ≈ −0.342, cos(−20°) ≈ 0.940cos doesn’t care about the negative; sin flips sign!
WE 4
Identify a graph from key features
A trig graph passes through (0, 0), reaches a maximum of 1 at x = 90°, returns to 0 at 180°, and a minimum of −1 at 270°. Which function is this?
Step 1: Check the y-intercept
Through (0, 0) → not cos (cos passes through (0, 1)).
Step 2: Check the maximum at 90°
Max of 1 at x = 90° matches sine perfectly.
Step 3: Check the rest
Returns to 0 at 180°, min at 270° — yes, that’s sin x.
y = sin x ✓tan also goes through (0,0) but has asymptotes — would never reach a max of 1
WE 5
Sketch y = cos θ and y = tan θ together (SME-style)
Sketch the graphs of y = cos θ and y = tan θ on the same set of axes for the interval −π ≤ θ ≤ 2π. Mark the key features clearly.
Step 1: Set up the axes (radians)
Mark θ-axis at −π, −π/2, 0, π/2, π, 3π/2, 2π. Draw horizontal dashed lines at y = 1 and y = −1.
Step 2: Sketch y = cos θ
Maximum at (0, 1), zero at π/2, minimum at π (= −1), zero at 3π/2, max at 2π. Same wave going leftwards.
Step 3: Sketch y = tan θ
Asymptotes at θ = ±π/2 and 3π/2. Curve passes through 0 at θ = 0 and θ = π. Goes up to ∞ near asymptotes from the left, down from −∞ on the right.
Step 4: Mark key features
Cos: max/min points labelled. Tan: asymptotes drawn as dashed vertical lines.
Both graphs sketched on same axes ✓drawing the cos wave first, then asymptotes, then tan branches is the cleanest order
💡 Top tips
Memorise the wave shapes — sin starts at 0, cos starts at 1, both peak at ±1. You’ll need to sketch them quickly under exam pressure.
Tan has asymptotes at ±90°, ±270°, etc. Always draw these as dashed vertical lines first, then fill in the branches.
Period of tan is 180°, NOT 360° — easy to forget. Sin and cos repeat every 360°; tan repeats every 180°.
Sin is odd, cos is even, tan is odd. sin(−x) = −sin x, cos(−x) = cos x, tan(−x) = −tan x.
Check your calculator mode (degrees vs radians) before drawing or evaluating.
For domains in radians, the key points are 0, π/2, π, 3π/2, 2π — these correspond to 0°, 90°, 180°, 270°, 360°.
Cos = sin shifted left by 90°. If you can sketch one, you can sketch the other.
⚠ Common mistakes
Wrong starting point. Sin starts at 0; cos starts at 1. Mixing these up is the #1 sketching error.
Forgetting tan’s asymptotes. Without them, the graph looks wrong — it must “break” at every ±90°.
Wrong period for tan. Tan repeats every 180° (π), not 360° (2π) like sin and cos.
Drawing tan as a wave. It’s not — it’s a curve that approaches infinity, breaks, and re-emerges. Definitely not symmetric like sine.
Treating cos as odd. Cos is even: cos(−x) = cos x. It’s only sin and tan that are odd.
Confusing radians and degrees. If the question is in radians, use π/2 not 90° on your axis labels.
Ignoring the domain. If the question asks about [−π, 2π], make sure your sketch actually covers that range.
These three graphs underpin every trig question on the IB syllabus — solving equations, transformations, modelling tides and Ferris wheels. Get fluent at sketching them by hand and the rest of the topic becomes much easier.
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