3.7 Circular functions

Many quantities in nature and technology repeat themselves at regular intervals: the height of the tide, the number of hours of daylight through a year, the height of a seat on a rotating wheel, the voltage in a mains electricity supply. The circular functions sine, cosine and tangent are the basic mathematical models of this kind of repeating behaviour. They grow out of the unit circle, where $\cos\theta$ and $\sin\theta$ are the coordinates of a point, and turn into functions of a real variable whose graphs are waves.

Stretching and translating the basic sine wave gives the family of functions $f(x) = a\sin(b(x + c)) + d$. Each of the four parameters controls one visible feature of the graph, so a graph or a set of data can be turned into an equation, and an equation into a graph, by reading those features.

From the unit circle to circular functions

On the unit circle, the point $P$ reached by turning through an angle $\theta$ anticlockwise from the positive $x$-axis has coordinates $(\cos\theta, \sin\theta)$. This definition works for any angle at all, not only for angles in a triangle: an angle larger than $2\pi$ simply takes $P$ round the circle more than once, and a negative angle takes it round clockwise. Angles are measured in radians unless degrees are stated (radians are introduced in 3.4).

Because every real number $x$ can be used as an angle in radians, $\sin x$ and $\cos x$ are defined for every $x \in \mathbb{R}$. Reading the angle as the input and a coordinate of $P$ as the output turns the unit circle into two functions.

The name "circular" comes from this construction; "trigonometric functions" means the same thing. As the angle $x$ increases from $0$, the point $P$ travels round the circle and its height above the horizontal axis rises from $0$ to $1$, falls back through $0$ to $-1$, and returns to $0$ when $x = 2\pi$. Plotting that height against $x$ unrolls the circle into the graph of $y = \sin x$.

The same construction with the $x$-coordinate of $P$ gives the graph of $y = \cos x$. The applet animates the point and draws each graph as it moves.

Periodic functions

After one full turn, $P$ is back where it started, so the values of $\sin x$ and $\cos x$ start to repeat. Adding $2\pi$ to the angle never changes the position of $P$:

Functions whose graphs repeat in this way are called periodic.

The period of $\sin x$ and of $\cos x$ is $2\pi$. Numbers such as $4\pi$ and $6\pi$ also satisfy $f(x + p) = f(x)$, but the period is the smallest positive one. Once one cycle of a periodic graph is known, the whole graph follows by copying that cycle to the left and right.

The graph of y = sin x

A table of exact values at multiples of $\frac{\pi}{6}$ and $\frac{\pi}{2}$ (from the unit circle, see 3.5) is enough to plot one cycle accurately.

(The decimals $\pm 0.866$ stand for $\pm\frac{\sqrt{3}}{2}$.) Joining the points with a smooth curve gives a wave that starts at the origin, rises to a maximum of $1$ at $x = \frac{\pi}{2}$, crosses the $x$-axis again at $x = \pi$, falls to a minimum of $-1$ at $x = \frac{3\pi}{2}$ and returns to $0$ at $x = 2\pi$. Every later cycle repeats this pattern.

The key features of $y = \sin x$:

• Domain: $x \in \mathbb{R}$.

• Range: $-1 \le \sin x \le 1$.

• Period: $2\pi$.

• Zeros (the $x$-intercepts): $x = 0, \pm\pi, \pm 2\pi, \dots$, that is $x = k\pi$ for every integer $k$.

• Maximum value $1$ at $x = \frac{\pi}{2}, \frac{\pi}{2} \pm 2\pi, \dots$

• y-intercept: $(0, 0)$.

The height of the wave is described by its amplitude.

The graph of y = cos x

Plotting the $x$-coordinate of $P$ against the angle gives a wave with exactly the same shape as the sine curve. It starts at its maximum, $\cos 0 = 1$, falls through $0$ at $x = \frac{\pi}{2}$ to the minimum $-1$ at $x = \pi$, and rises back through $0$ at $x = \frac{3\pi}{2}$ to $1$ at $x = 2\pi$.

The key features of $y = \cos x$:

• Domain $x \in \mathbb{R}$; range $-1 \le \cos x \le 1$; period $2\pi$; amplitude $1$.

• Zeros: $x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \dots$

• Maximum $1$ at $x = 0, \pm 2\pi, \dots$; minimum $-1$ at $x = \pm\pi, \pm 3\pi, \dots$

• y-intercept: $(0, 1)$.

• The graph is symmetrical about the $y$-axis.

Sliding the sine curve $\frac{\pi}{2}$ units to the left lands it exactly on the cosine curve, so $\cos x = \sin\left(x + \frac{\pi}{2}\right)$ for every $x$. Both curves, and any curve obtained from them by stretches and translations, are called sinusoidal.

The graph of y = tan x

The third circular function is defined from the other two:

On the unit circle, $\tan x$ is the gradient of the radius $OP$. The quotient is undefined wherever $\cos x = 0$, that is at $x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \dots$ As $x$ approaches $\frac{\pi}{2}$ from below, $\sin x$ approaches $1$ while $\cos x$ approaches $0$ through positive values, so $\tan x$ grows without limit. Just after $\frac{\pi}{2}$, $\cos x$ is small and negative and $\tan x$ is very large and negative. The graph therefore has a vertical asymptote at each zero of $\cos x$.

The key features of $y = \tan x$:

• Domain: $x \in \mathbb{R}$, $x \neq \frac{\pi}{2} + k\pi$ for every integer $k$.

• Range: $\tan x \in \mathbb{R}$ (every real value is taken).

• Period: $\pi$. The point $P$ at angle $x + \pi$ is the reflection of $P$ through the origin, so both coordinates change sign and their quotient does not change.

• Zeros: $x = k\pi$, the same as the zeros of $\sin x$.

• Vertical asymptotes: $x = \frac{\pi}{2} + k\pi$, at the zeros of $\cos x$.

• No amplitude, because the graph has no maximum or minimum value. The term amplitude applies only to sine and cosine curves.

The three basic circular functions compare as follows.

Transformations of circular functions

The general rules for transforming graphs (2.11) apply to circular functions just as they apply to any other function. Because sinusoidal graphs repeat, each transformation has a direct effect on a named feature: the height of the wave, its length, its position along the $x$-axis or its central level.

Vertical stretch: the amplitude

Multiplying the function by a constant $a$ stretches the graph vertically with scale factor $a$. Every $y$-value is multiplied by $a$, so $y = a\sin x$ oscillates between $-|a|$ and $|a|$ and has amplitude $|a|$. When $a$ is negative, the graph is also reflected in the $x$-axis: $y = -2\sin x$ starts at the origin going down, not up. The amplitude is a distance, so it is never negative.

Horizontal stretch: the period

In $y = \sin(bx)$ with $b > 0$, the input to the sine function is $bx$. One cycle of $\sin$ is completed as its input runs from $0$ to $2\pi$, and $bx$ runs from $0$ to $2\pi$ when $x$ runs from $0$ to $\frac{2\pi}{b}$. So the graph is stretched horizontally with scale factor $\frac{1}{b}$, and

A value $b > 1$ squeezes more cycles into each interval of length $2\pi$; a value $0 < b < 1$ spreads them out. The number $b$ is the number of complete cycles in an interval of length $2\pi$.

Horizontal translation: the horizontal shift

Replacing $x$ by $x + c$ translates the graph horizontally by $-c$: $y = \sin(x + c)$ is the sine curve moved $c$ units to the left when $c > 0$, and $|c|$ units to the right when $c < 0$. The graph of $y = \sin\left(x - \frac{\pi}{3}\right)$ is the sine curve moved $\frac{\pi}{3}$ to the right, so its first upward zero is at $x = \frac{\pi}{3}$ instead of $0$. The amount $-c$ is called the horizontal shift (also called the phase shift).

When there is also a horizontal stretch, the bracket must be written with $b$ taken out as a factor, $\sin(b(x + c))$, before the shift can be read. The order matters: $\sin(2x + \pi)$ is $\sin\left(2\left(x + \frac{\pi}{2}\right)\right)$, a shift of $\frac{\pi}{2}$ to the left, not $\pi$.

Vertical translation: the principal axis

Adding a constant $d$ translates the graph $d$ units vertically. The wave $y = \sin x + d$ oscillates about the line $y = d$ instead of the $x$-axis.

The general sinusoidal function

Combining the four transformations gives the general sinusoidal function.

Each parameter controls one feature, and the others do not affect it.

Two features follow straight from these: the maximum value is $d + |a|$ and the minimum value is $d - |a|$, so the range is $d - |a| \le f(x) \le d + |a|$.

Describing a sequence of transformations

To describe how $y = \sin x$ (or $y = \cos x$) is mapped onto $y = a\sin(b(x + c)) + d$, list the horizontal transformations and the vertical transformations. Horizontal changes act on $x$ and vertical changes act on $y$, so the two groups do not interfere with each other. Within each group, the order matters:

• Horizontally: the stretch with scale factor $\frac{1}{b}$, then the translation by $-c$. (Doing the translation first would give $\sin(bx + c)$, which is a different function.)

• Vertically: the stretch with scale factor $a$, then the translation by $d$. (Doing the translation first would give $a(\sin x + d) = a\sin x + ad$.)

A translation is often written as a vector, such as $\begin{pmatrix} -c \\ 0 \end{pmatrix}$ for a horizontal translation by $-c$.

Sketching sinusoidal graphs

A sinusoidal graph can be sketched quickly from five key points per cycle. For a sine or cosine curve, the maxima, minima and crossings of the principal axis are spaced exactly a quarter of a period apart.

1. Find the amplitude $|a|$, the period $\frac{2\pi}{b}$, the shift $-c$ and the principal axis $y = d$.

2. Draw the principal axis and the lines $y = d + |a|$ and $y = d - |a|$ as light guides.

3. Find one starting point. For $a\sin(b(x + c)) + d$ with $a > 0$ the curve crosses the principal axis going up at $x = -c$

4. Step forwards and backwards in quarter periods, marking the next maximum, crossing, minimum and crossing in turn.

5. Join the points with a smooth wave, rounded at the turning points, and cut it off at the ends of the given domain. Label the axis intercepts and the end points.

Transformed tangent functions

The same transformations apply to $y = \tan x$, with two differences: there is no amplitude (the factor $a$ only makes the branches steeper or shallower), and the period of $\tan(b(x + c))$ is $\frac{\pi}{b}$. The vertical asymptotes move with the graph: they are where the input to $\tan$ equals $\frac{\pi}{2} + k\pi$.

Finding the equation from a graph

Reading a sinusoidal graph backwards gives its equation. The four parameters are found one at a time, each from the feature it controls.

1. Principal axis: $d = \dfrac{\text{max} + \text{min}}{2}$.

2. Amplitude: $|a| = \dfrac{\text{max} - \text{min}}{2}$.

3. Period: measure from one maximum to the next (or between any two matching points), then $b = \dfrac{2\pi}{\text{period}}$.

4. Shift: choose a starting point. For a sine model with $a > 0$, use a point where the graph crosses the principal axis going up; for a cosine model with $a > 0$, use a maximum. If that point has $x$-coordinate $x_0$

The answer is not unique. Because the graph repeats, any starting point one period further on gives an equally valid value of $c$, and the same curve can be written with sine or with cosine. A question normally restricts the parameters (for example $a > 0$ and $0 \le c < \pi$) so that only one answer fits. These different equations are equivalent: they describe the same function.

When the given information is a list of facts rather than a graph, the same four steps apply. A quick sketch of the facts helps to see where the maxima and minima are.

Sinusoidal models of real data

A periodic quantity that rises and falls smoothly between fixed limits can often be modelled by a sinusoidal function of time $t$. The modelling process uses the same four steps as finding an equation from a graph, with data in place of a drawn curve:

• the principal axis $d$ is the mean of the highest and lowest values;

• the amplitude $|a|$ is half their difference;

• the period comes from the time for one full cycle, and $b = \frac{2\pi}{\text{period}}$;

• the shift comes from the time of a known maximum (cosine) or of a value on the principal axis on the way up (sine).

A model is a simplification, so a good model is checked against data that were not used to build it. The difference between the model and reality shows how far the model can be trusted, an idea of modelling and validity that runs through the whole course.

Tides

The height of the sea at a port rises and falls about twice a day. The tide table for Boston, Massachusetts, on 3 June 2025 lists the following high and low waters (heights above the local reference level, Mean Lower Low Water).

Tide predictions: NOAA Tides and Currents, station 8443970 (Boston, MA), 3 June 2025.

Daylight hours

The number of hours of daylight in a city changes through the year with a period of one year. Away from the equator, the change is large: in London the longest day, 21 June, has 16 hours 38 minutes of daylight, and the shortest, 21 December, has 7 hours 49 minutes (sunrisesunset.io, 2026). The curve of daylight against the day of the year is close to sinusoidal.

The London Eye

A Ferris wheel turns at a constant rate, so the height of a capsule is a sinusoidal function of time. The London Eye has a diameter of $120$ m and makes one revolution in about $30$ minutes. In a simplified model, the lowest point of a capsule's path is taken as ground level ($h = 0$) and a capsule starts at the lowest point at $t = 0$.

How good is a sinusoidal model?

A sinusoidal model assumes a fixed amplitude, a fixed period and a perfectly smooth, symmetrical rise and fall. Real data often depart from this. Tidal ranges grow and shrink over each lunar month; temperatures rise faster in the morning than they fall in the evening; daylight curves flatten near the solstices more than a sine curve does. Before relying on a model, compare it with data that were not used to build it, look at whether the errors follow a pattern, and restrict predictions to the time span the data cover. A model built from one day of tides is reliable for a day or two, not for a month.

Exam-style questions on circular functions

The first question is in the style of a calculator-free paper; the second allows a GDC.

Self-check

• To do:

  I can sketch $y = \sin x$, $y = \cos x$ and $y = \tan x$ and state their domain, range, period, zeros and asymptotes.

• To do:

  I can explain why the circular functions are periodic, using the unit circle.

• To do:

  I can find the amplitude, period, horizontal shift and principal axis of $a\sin(b(x + c)) + d$ and $a\cos(b(x + c)) + d$

• To do:

  I can take out the factor $b$ before reading a horizontal shift, as in $\sin(2x + \pi) = \sin\left(2\left(x + \frac{\pi}{2}\right)\right)$

• To do:

  I can describe the sequence of transformations that maps $y = \sin x$ onto a given sinusoidal function.

• To do:

  I can sketch a transformed sine, cosine or tangent graph using key points a quarter period apart.

• To do:

  I can find the equation of a sinusoidal function from its graph or from its maximum and minimum points.

• To do:

  I can build a sinusoidal model from real data, use it to make predictions and comment on its validity.

where $t$ is the time in hours after midnight.

1. (a)

   Find the depth of water at 07:00.

   [2]2 marks

   Markscheme (a)

   $d(7) = 7 + 4\sin\left(\dfrac{7\pi}{6}\right)$ (M1)

   $= 7 + 4 \times \left(-\dfrac{1}{2}\right) = 5$ (m) A1

2. (b)

   Find the exact depth of water at 10:00.

   [2]2 marks

   Markscheme (b)

   $d(10) = 7 + 4\sin\left(\dfrac{5\pi}{3}\right)$

3. (c)

   Find the maximum depth of water and the time at which it occurs.

   [2]2 marks

   Markscheme (c)

   maximum depth $= 11$ (m) A1

   $\sin\left(\dfrac{\pi t}{6}\right) = 1 \Rightarrow \dfrac{\pi t}{6} = \dfrac{\pi}{2}$

(b) $3\pi = 2\pi + \pi$, so $\cos(3\pi) = \cos\pi = -1$.

(c) $-\frac{\pi}{2}$ is a quarter turn clockwise, which puts $P$ at $(0, -1)$. The $y$-coordinate gives $\sin\left(-\frac{\pi}{2}\right) = -1$.

(d) $\frac{13\pi}{3} = 4\pi + \frac{\pi}{3}$, so $\cos\left(\frac{13\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$.

(a) Amplitude $= |a| = 3$.

(b) Period $= \frac{2\pi}{b} = \frac{2\pi}{2} = \pi$.

(c) The principal axis is $y = d$, that is $y = 1$.

(d) The maximum value is $d + |a| = 4$ and the minimum value is $d - |a| = -2$, so the range is $-2 \le f(x) \le 4$.

The graph is also the sine curve moved $\frac{\pi}{4}$ to the right; that shift affects none of these four answers.

So $a = -2$, $b = 3$, $c = \frac{\pi}{6}$ and $d = 5$.

Horizontal transformations, in this order:

• a horizontal stretch with scale factor $\frac{1}{3}$ (parallel to the $x$-axis), giving $y = \sin 3x$;

• a translation $\frac{\pi}{6}$ to the left, by the vector $\begin{pmatrix} -\frac{\pi}{6} \\ 0 \end{pmatrix}$, giving $y = \sin\left(3\left(x + \frac{\pi}{6}\right)\right)$.

Vertical transformations, in this order:

• a vertical stretch with scale factor $2$ and a reflection in the $x$-axis (together, a vertical stretch with scale factor $-2$);

• a translation $5$ units up, by the vector $\begin{pmatrix} 0 \\ 5 \end{pmatrix}$.

The amplitude is $|-2| = 2$ and the principal axis is $y = 5$, so the range is $5 - 2 \le y \le 5 + 2$, that is $3 \le y \le 7$.

• Amplitude $2$; period $\frac{2\pi}{1/2} = 4\pi$, so the domain covers exactly one cycle.

• Principal axis $y = 1$; maximum value $1 + 2 = 3$; minimum value $1 - 2 = -1$.

• Since $a < 0$, the cosine curve is reflected: it starts at a minimum at $x = 0$.

A quarter period is $\pi$, which gives the key points:

For the $x$-intercepts, set $y = 0$: $\cos\left(\frac{x}{2}\right) = \frac{1}{2}$. With $0 \le \frac{x}{2} \le 2\pi$, $\frac{x}{2} = \frac{\pi}{3}$ or $\frac{5\pi}{3}$, so $x = \frac{2\pi}{3}$ or $x = \frac{10\pi}{3}$ (solving such equations is covered in 3.8).

Zeros occur where $\frac{x}{2} = k\pi$, that is $x = 2k\pi$. Inside the open interval the only zero is $x = 0$ (the values $\pm 2\pi$ are excluded).

At $x = \frac{\pi}{2}$, $y = 2\tan\frac{\pi}{4} = 2$, which helps fix the steepness of the central branch.

Period. From a minimum to the next maximum is half a period, so

Shift. With $a > 0$, a sine curve crosses the principal axis going up a quarter period after a minimum. A quarter period is $\frac{1}{4} \times \frac{2\pi}{3} = \frac{\pi}{6}$, so the curve crosses $y = 2$ going up at $x = \frac{\pi}{6}$. This is the point where $b(x + c) = 0$, so

This lies in the interval $-\frac{\pi}{3} < c \le 0$, so

Check. $f(0) = 3\sin\left(-\frac{\pi}{2}\right) + 2 = -3 + 2 = -1$ and $f\left(\frac{\pi}{3}\right) = 3\sin\left(\frac{\pi}{2}\right) + 2 = 5$. Both points fit.

Without the restriction on $c$, the value $c = -\frac{\pi}{6} + \frac{2\pi}{3} = \frac{\pi}{2}$ would also work, and the same curve can be written as $f(x) = -3\cos(3x) + 2$, a reflected cosine curve that has its minimum on the $y$-axis.

A cosine curve with $p > 0$ has a maximum at $x = 0$, which fits the given point with no shift. From the maximum to the next minimum is half a period, so the period is $2 \times \frac{\pi}{4} = \frac{\pi}{2}$ and

So $g(x) = 3\cos(4x) + 4$.

(b) $g\left(\frac{\pi}{12}\right) = 3\cos\left(\frac{4\pi}{12}\right) + 4 = 3\cos\left(\frac{\pi}{3}\right) + 4 = 3 \times \frac{1}{2} + 4 = \frac{11}{2}$.

From high water to low water is half a period: $12.117 - 5.817 = 6.3$ hours, so the period is $12.6$ hours and

A sine model with $a > 0$ has a maximum where $b(t + c) = \frac{\pi}{2}$, that is a quarter period after $t = -c$. A quarter period is $\frac{12.6}{4} = 3.15$ hours, so

To 3 significant figures, the model is

(b) At 09:00, $t = 9$. Using the unrounded values,

The model estimates a height of about $1.56$ m, just below the principal axis, as expected half way between high and low water.

(c) At 18:29, $t \approx 18.483$, the model gives $h \approx 2.84$ m against the predicted $2.78$ m: a difference of about $0.06$ m, and the time of high water is matched almost exactly. The model fits well over one day. Real tides have successive highs (and lows) of different heights, because the Moon and the Sun do not pull in step, so a single sine curve with a fixed amplitude becomes less accurate over several days.

The period is $365$ days, so $b = \frac{2\pi}{365} \approx 0.0172$.

A cosine model with $a > 0$ has its maximum where $t + c = 0$, and the maximum is at $t = 172$, so $c = -172$:

(b) A sine curve reaches its maximum a quarter period after it crosses the principal axis going up. A quarter of $365$ days is $91.25$ days, so the sine curve crosses upwards at $t = 172 - 91.25 = 80.75$:

Here $c = -80.75$. The value $t = 80.75$ is about 21 March, the spring equinox, when day and night are roughly equal (the model gives $d \approx 12.2$ hours).

(c) Using the unrounded values,

The model gives about $10.0$ hours (10 hours 0 minutes), while the true value is 10 hours 23 minutes, a difference of about $4\%$. The model captures the yearly pattern well, but it is not exact: the real curve is not perfectly sinusoidal, and the published times count daylight from the first to the last moment the edge of the Sun is visible, so even the equinox has a little more than 12 hours of daylight.

which is the given result.

(b) $h(5) = 60 - 60\cos\left(\frac{\pi}{3}\right) = 60 - 60 \times \frac{1}{2} = 30$ m.

$h(25) = 60 - 60\cos\left(\frac{5\pi}{3}\right) = 60 - 30 = 30$ m. The capsule is at the same height $5$ minutes after boarding and $5$ minutes before returning to the bottom, on opposite sides of the wheel.

(c) The amplitude is $60$, the principal axis is $h = 60$ and $b = \frac{\pi}{15}$. A sine curve with $a > 0$ crosses its principal axis going up a quarter period before its maximum. The maximum, $120$ m, is at $t = 15$, and a quarter period is $7.5$ minutes, so the upward crossing is at $t = 7.5$ and $c = -7.5$:

Check: $h(0) = 60\sin\left(-\frac{\pi}{2}\right) + 60 = 0$.

(c) The maximum occurs where $\sin\left(2\left(x - \frac{\pi}{6}\right)\right) = 1$, that is where $2\left(x - \frac{\pi}{6}\right) = \frac{\pi}{2}$: