Graphs of trigonometric functions

Sine, cosine and tangent repeat with a fixed period, and their reciprocals sec, cosec and cot have asymptotes wherever the function they invert is zero. x is in radians throughout.

Every value in the key-features tables is checked by script against the Grapher's own calculations (roots, turning points, asymptotes, domain and range).

Graph of y = sin x

The sine wave (x in radians): it oscillates between −1 and 1 and repeats every 2π. It passes through the origin.

Open y = sin x in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	−1 ≤ y ≤ 1
x-intercepts	x = kπ for every integer k (… −π, 0, π, 2π …)
y-intercept	(0, 0)
Turning points	maxima (π/2 + 2kπ, 1); minima (−π/2 + 2kπ, −1)
Asymptotes	none
Period	2π
Amplitude	1
Symmetry	odd function: f(−x) = −f(x), rotational symmetry of order 2 about the origin

How to sketch it

1. Mark the x-intercepts x = kπ for every integer k (… −π, 0, π, 2π …), the y-intercept (0, 0), the maxima (π/2 + 2kπ, 1); minima (−π/2 + 2kπ, −1).
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: −1 ≤ y ≤ 1. Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.
5. Draw one period (2π long) and repeat it.

Graphs made from this one: y = cos x, y = sin 2x, y = 2 sin x + 1, y = cosec x, y = arcsin x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = cos x

The cosine wave (x in radians): the sine wave shifted π/2 to the left, so it starts at its maximum, (0, 1).

Open y = cos x in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	−1 ≤ y ≤ 1
x-intercepts	x = π/2 + kπ for every integer k
y-intercept	(0, 1)
Turning points	maxima (2kπ, 1); minima (π + 2kπ, −1)
Asymptotes	none
Period	2π
Amplitude	1
Symmetry	even function: f(−x) = f(x), symmetric about the y-axis

How to sketch it

1. Mark the x-intercepts x = π/2 + kπ for every integer k, the y-intercept (0, 1), the maxima (2kπ, 1); minima (π + 2kπ, −1).
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: −1 ≤ y ≤ 1. Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and reflect it in the y-axis.
5. Draw one period (2π long) and repeat it.

y = cos x is y = sin x translated π/2 to the left: cos x = sin(x + π/2): see the graph of y = sin x. Graphs made from this one: y = sec x, y = arccos x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = tan x

The tangent graph (x in radians): it repeats every π, with vertical asymptotes where cos x = 0.

Open y = tan x in the Grapher

Key features

Domain	x ≠ π/2 + kπ (k an integer)
Range	all real numbers (y ∈ ℝ)
x-intercepts	x = kπ for every integer k
y-intercept	(0, 0)
Point of inflection	(kπ, 0)
Asymptotes	vertical: x = π/2 + kπ (… −π/2, π/2, 3π/2 …)
Period	π
Symmetry	odd function: f(−x) = −f(x), rotational symmetry of order 2 about the origin

How to sketch it

1. Draw the asymptotes first as dashed lines: x = π/2 + kπ (… −π/2, π/2, 3π/2 …).
2. Mark the x-intercepts x = kπ for every integer k, the y-intercept (0, 0).
3. Only draw the curve where it is defined: x ≠ π/2 + kπ (k an integer).
4. Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.
5. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.
6. Draw one period (π long) and repeat it.

Graphs made from this one: y = cot x, y = arctan x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = sin 2x

The sine wave squashed horizontally by a factor of 2, so it repeats twice as often: period π.

Open y = sin 2x in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	−1 ≤ y ≤ 1
x-intercepts	x = kπ/2 for every integer k
y-intercept	(0, 0)
Turning points	maxima (π/4 + kπ, 1); minima (−π/4 + kπ, −1)
Asymptotes	none
Period	π
Amplitude	1
Symmetry	odd function: f(−x) = −f(x), rotational symmetry of order 2 about the origin

How to sketch it

1. Mark the x-intercepts x = kπ/2 for every integer k, the y-intercept (0, 0), the maxima (π/4 + kπ, 1); minima (−π/4 + kπ, −1).
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: −1 ≤ y ≤ 1. Your sketch should not go anywhere else.
4. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.
5. Draw one period (π long) and repeat it.

y = sin 2x is y = sin x stretched horizontally by scale factor 1/2: see the graph of y = sin x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = 2 sin x + 1

A sine wave with amplitude 2 and principal axis y = 1, so it oscillates between −1 and 3.

Open y = 2 sin x + 1 in the Grapher

Key features

Domain	all real numbers (x ∈ ℝ)
Range	−1 ≤ y ≤ 3
x-intercepts	where sin x = −1/2: x = −π/6 + 2kπ and x = 7π/6 + 2kπ
y-intercept	(0, 1)
Turning points	maxima (π/2 + 2kπ, 3); minima (−π/2 + 2kπ, −1)
Asymptotes	none
Period	2π
Amplitude	2
Principal axis	y = 1
Symmetry	neither even nor odd

How to sketch it

1. Mark the x-intercepts where sin x = −1/2: x = −π/6 + 2kπ and x = 7π/6 + 2kπ, the y-intercept (0, 1), the maxima (π/2 + 2kπ, 3); minima (−π/2 + 2kπ, −1).
2. The domain is every real number, so the curve continues off both sides of your axes.
3. Check the range: −1 ≤ y ≤ 3. Your sketch should not go anywhere else.
4. Draw one period (2π long) and repeat it.

y = 2 sin x + 1 is y = sin x stretched vertically by scale factor 2, then translated up 1: see the graph of y = sin x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = sec x

sec x = 1/cos x: U-shaped and ∩-shaped branches between vertical asymptotes where cos x = 0. It repeats every 2π.

Open y = sec x in the Grapher

Key features

Domain	x ≠ π/2 + kπ (k an integer)
Range	y ≤ −1 or y ≥ 1
x-intercepts	none
y-intercept	(0, 1)
Turning points	local minima (2kπ, 1); local maxima (π + 2kπ, −1)
Asymptotes	vertical: x = π/2 + kπ
Period	2π
Symmetry	even function: f(−x) = f(x), symmetric about the y-axis

How to sketch it

1. Draw the asymptotes first as dashed lines: x = π/2 + kπ.
2. Mark the y-intercept (0, 1), the local minima (2kπ, 1); local maxima (π + 2kπ, −1).
3. Only draw the curve where it is defined: x ≠ π/2 + kπ (k an integer).
4. Check the range: y ≤ −1 or y ≥ 1. Your sketch should not go anywhere else.
5. Use the symmetry: draw the right-hand half and reflect it in the y-axis.
6. Draw one period (2π long) and repeat it.

y = sec x is the reciprocal of y = cos x: see the graph of y = cos x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = cosec x

cosec x = 1/sin x: branches between vertical asymptotes where sin x = 0, at every multiple of π. It repeats every 2π.

Open y = cosec x in the Grapher

Key features

Domain	x ≠ kπ (k an integer)
Range	y ≤ −1 or y ≥ 1
x-intercepts	none
y-intercept	none (sin 0 = 0, so x = 0 is not in the domain)
Turning points	local minima (π/2 + 2kπ, 1); local maxima (−π/2 + 2kπ, −1)
Asymptotes	vertical: x = kπ (… −π, 0, π …)
Period	2π
Symmetry	odd function: f(−x) = −f(x), rotational symmetry of order 2 about the origin

How to sketch it

1. Draw the asymptotes first as dashed lines: x = kπ (… −π, 0, π …).
2. Mark the local minima (π/2 + 2kπ, 1); local maxima (−π/2 + 2kπ, −1).
3. Only draw the curve where it is defined: x ≠ kπ (k an integer).
4. Check the range: y ≤ −1 or y ≥ 1. Your sketch should not go anywhere else.
5. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.
6. Draw one period (2π long) and repeat it.

y = cosec x is the reciprocal of y = sin x: see the graph of y = sin x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

Graph of y = cot x

cot x = cos x / sin x: decreasing branches between vertical asymptotes at multiples of π. It repeats every π.

Open y = cot x in the Grapher

Key features

Domain	x ≠ kπ (k an integer)
Range	all real numbers (y ∈ ℝ)
x-intercepts	x = π/2 + kπ
y-intercept	none (x = 0 is not in the domain)
Asymptotes	vertical: x = kπ (… −π, 0, π …)
Period	π
Symmetry	odd function: f(−x) = −f(x), rotational symmetry of order 2 about the origin

How to sketch it

1. Draw the asymptotes first as dashed lines: x = kπ (… −π, 0, π …).
2. Mark the x-intercepts x = π/2 + kπ.
3. Only draw the curve where it is defined: x ≠ kπ (k an integer).
4. Check the range: all real numbers (y ∈ ℝ). Your sketch should not go anywhere else.
5. Use the symmetry: draw the right-hand half and rotate it 180° about the origin.
6. Draw one period (π long) and repeat it.

y = cot x is the reciprocal of y = tan x: see the graph of y = tan x. Transform it with sliders for a, b, c and d in y = af(b(x − c)) + d.

The Grapher is our own free tool (graph.mathrevision.co.uk), built for IB, A Level and IGCSE students. It is not an exam calculator: practise on your own GDC for calculator papers.