IB Maths AI HL Trig Identities & Equations Paper 1 & 2 Sin, cos, tan graphs ~8 min read

Graphs of Trigonometric Functions

The graphs of sin, cos and tan are all periodic, but they have distinct shapes. Sin and cos are smooth waves with amplitude 1 and period 360° (2π); they differ only by a horizontal shift. Tan repeats every 180° (π), with vertical asymptotes at the points where cos = 0. Knowing each graph’s key values, period, range, and symmetry is the foundation for everything that follows.

📘 What you need to know

y = sin x: domain ℝ, range [−1, 1], period 360°/2π, passes through (0, 0), odd function: sin(−x) = −sin x.
y = cos x: domain ℝ, range [−1, 1], period 360°/2π, passes through (0, 1), even function: cos(−x) = cos x.
Sin and cos are shifts of each other: cos x = sin(x+90°), so the cos graph is the sin graph shifted 90° to the left.
y = tan x: period 180°/π, range ℝ, vertical asymptotes at x = 90°+180°k (i.e. π/2+πk); passes through (0, 0) and is odd.
Sin and cos key values: at multiples of 90° they take values from { −1, 0, 1 }. Tan is 0 at multiples of 180° and undefined at odd multiples of 90°.
Symmetry: sin has rotational symmetry about the origin; cos has reflective symmetry about the y-axis; tan has rotational symmetry about the origin and about every asymptote.

The three graphs and their key features

Sin and cos look almost identical — both smooth waves bounded between −1 and 1, with period 360° (2π). The difference is where they cross the y-axis: sin at 0, cos at 1. They are horizontal shifts of each other (cos x is sin x shifted left by 90°). Tan is completely different: it has no maximum or minimum, repeats every 180° (π), and has vertical asymptotes wherever cos = 0 (because tan = sin/cos blows up there).

Top: sin (red) and cos (teal) on the same axes — identical waves shifted by π/2. Bottom: tan (teal) with three asymptotes at x = ±π/2, 3π/2 (dashed red), repeating every π and passing through 0 at multiples of π.

Quick-reference sin x: period 360°/2π · range [−1, 1] · odd · through (0, 0) cos x: period 360°/2π · range [−1, 1] · even · through (0, 1) tan x: period 180°/π · range ℝ · odd · asymptotes at 90°+180°k

Reading values and sketching

For values at multiples of 90° (or π/2), don’t reach for a calculator — read them off the graph. Sin and cos cycle through {0, 1, 0, −1} repeating, just starting from different points. To sketch over a given interval, first set the x-axis scale: use multiples of 90° (or π/2) and check whether the question is in degrees or radians. Then plot the values at these multiples, mark any asymptotes (for tan), and join with smooth curves matching the standard shape.

Key-value cycles: sin x at x = 0°, 90°, 180°, 270°, 360° gives 0, 1, 0, −1, 0; cos x at the same gives 1, 0, −1, 0, 1. Tan x at multiples of 180° is 0; at odd multiples of 90° it’s undefined.

🧭 Recipe — sketch a trig graph

Check units: degrees or radians? Match the calculator and your x-axis labels.
Label the x-axis with multiples of 90° (or π/2) covering the whole given interval.
Label the y-axis: [−1, 1] for sin/cos; show enough range for tan to make the asymptotes clear.
Identify key values at each multiple of 90° using the standard cycle. Mark asymptotes for tan.
Join with smooth curves: sin/cos as waves through zeros and turning points; tan as four curves climbing between asymptotes.

Worked examples

WE 1

Key values — read off the graphs

Without using a calculator, write down the value of:
(a) sin(180°), (b) cos(270°), (c) tan(360°), (d) sin(−90°), (e) cos(0°), (f) tan(180°).

use the standard graphs (a) sin(180°) = 0 (b) cos(270°) = 0 (c) tan(360°) = 0 (d) sin(−90°) = −1 (e) cos(0°) = 1 (f) tan(180°) = 0 all six values done from memory sin/cos cycle through {0, 1, 0, −1}; tan is 0 at all multiples of 180°.

WE 2

Properties of y = sin x

The function y = sin x is given for x in degrees. State:
(a) the period; (b) the amplitude; (c) the range; (d) the coordinates of the first maximum point at x > 0.

read from the graph (a) period = 360° (b) amplitude = 1 (c) range = [−1, 1] (d) first maximum: (90°, 1) amplitude = half the distance from max to min = (1 − (−1))/2 = 1.

WE 3

Use odd/even symmetry of the graphs

Use the symmetries of the trig graphs to evaluate, without a calculator:
(a) sin(−30°), given that sin(30°) = 0.5;
(b) cos(−60°), given that cos(60°) = 0.5;
(c) tan(−45°), given that tan(45°) = 1.

(a) sin is odd: sin(−x) = −sin(x) sin(−30°) = −sin(30°) = −0.5 −0.5 (b) cos is even: cos(−x) = cos(x) cos(−60°) = cos(60°) = 0.5 0.5 (c) tan is odd: tan(−x) = −tan(x) tan(−45°) = −tan(45°) = −1 −1 sin has rotational symmetry about the origin; cos is symmetric about the y-axis.

WE 4

Read solutions of cos x = 0 from the graph

By referring to the graph of y = cos x, find all values of x in −360° ≤ x ≤ 360° for which cos x = 0.

cos x = 0 at odd multiples of 90° ±90°, ±270°, ±450°, … keep only those in [−360°, 360°] x = −270°, −90°, 90°, 270° four solutions in a 720° window — consistent with cos crossing zero twice per period.

WE 5

Asymptotes and intercepts of y = tan x

Consider the graph of y = tan x in the interval −π ≤ x ≤ 2π (radians).
(a) State the equations of all vertical asymptotes.
(b) State the x-coordinates of all points where the graph crosses the x-axis.

(a) asymptotes where cos x = 0 x = π/2 + kπ for integer k in [−π, 2π]: k = −1, 0, 1 x = −π/2, π/2, 3π/2 (b) x-intercepts where tan x = 0 (sin x = 0) x = kπ for integer k x = −π, 0, π, 2π asymptotes and intercepts alternate every π/2 along the x-axis.

WE 6

Compare sin and cos graphs

By sketching y = sin x and y = cos x on the same axes for 0° ≤ x ≤ 360°:
(a) State the x-coordinates of the points where the graphs intersect.
(b) State the values of x for which sin x > cos x.

(a) intersections: sin x = cos x divide by cos x → tan x = 1 x = 45°, 225° in [0°, 360°] intersections at x = 45°, 225° (b) sin > cos: check a test point between intersections at x = 90°: sin = 1, cos = 0 → sin > cos ✓ at x = 270°: sin = −1, cos = 0 → sin < cos sin x > cos x for 45° < x < 225° between intersections, the relative order can only switch at an intersection — one test point per region settles it.

💡 Top tips

Memorise both standard shapes — sin (starts at 0, up first) and cos (starts at 1, down first) — and the tan branches with asymptotes.
Cycle table: sin (0,1,0,−1) and cos (1,0,−1,0) at 0°, 90°, 180°, 270° — recite these to read any value at a multiple of 90°.
Odd/even shortcuts: sin(−x) = −sin x, cos(−x) = cos x, tan(−x) = −tan x — saves time on negative-angle questions.
Tan asymptotes occur where cos = 0 — so once you know the cos graph, you know where tan blows up.
For intersections of sin and cos, divide by cos to get tan x = 1 and solve from the tan graph.

⚠ Common mistakes

Confusing sin and cos shapes — remember sin passes through (0, 0), cos passes through (0, 1).
Wrong period for tan — tan repeats every 180° (π), not 360°.
Missing asymptotes when sketching tan — draw the vertical dashed lines first, then the branches between them.
Forgetting that “amplitude” is half the peak-to-trough distance, not the maximum value (the two coincide for plain sin/cos because the midline is 0).
Mixing units: an interval given in radians must use π/2 labels; an interval in degrees uses 90° labels.

Next up — Solving Equations Using Trigonometric Graphs. The graphs you’ve sketched here become the tool for finding every solution of sin x = k, cos x = k, tan x = k in an interval.

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