The unit circle

In the study of circular functions, the unit circle plays a central role in linking angles with trigonometric values. By defining sine, cosine, and tangent in terms of coordinates on a circle of radius 1, we gain a deeper understanding of how these functions behave over different angle measures. The unit circle helps visualise key concepts such as periodicity, symmetry, and angle relationships in both degrees and radians, forming a foundation for solving trigonometric equations and modelling real-world phenomena involving cycles or waves.

Use this page to revise the following concepts of the unit circle:

Structure of the Unit Circle

The unit circle is defined as a circle of radius 1, centred on the origin. The circumference around a circle is given by \(C = 2\pi{r}\). As the radius of the unit circle is 1, the circumference is \(C = 2\pi\) units.

Circle of radius one on a cartesian plane labelled x, y. The circumference of the circle intersects axes at (0,-1), (0,1), (-1,0) and (1,0)

We often need to discuss the direction and degree of rotation and angles around the unit circle.  Moving in an anticlockwise direction, for example, moving from point \(A\) to point \(B\), is considered a positive rotation, while moving in a clockwise direction, from point \(A\) to point \(D\),  is considered a negative rotation. We can also describe the angles formed between a radius line of the unit circle and the axes. These angles and rotations can be measured in degrees or radians.

Degrees and Radians

The unit circle has a distance around the circumference of \(C =2\pi\) units.

A graph of the unit circle. The circle is centred at the origin (0,0) with a radius of 1. The x-axis and y-axis are marked with a grid, and the circle intersects the axes at A(1,0), B(-1,0), C(0,1), and D(0,-1). A point P is marked in the first quadrant, and is distance r along the circumference of the unit circle from A. The angle formed between the x-axis and the radius to point P is marked with the angle theta. Below the graph is listed theta=57.296 degrees=1 radian

The distance around the circumference from \(A\) to \(P\) is 1 unit. The angle that is formed at the origin of the circle that corresponds to this distance is called a radian. By this metric, we can also see the the angle from A all the way around and back to A is \(2\pi\) radians.
The unit of measurement for radian is \(\text{rad}\). Often, the unit is omitted; for example \(\frac{\pi}{4}\) is assumed to refer to \(\frac{\pi}{4}\) radians. Angles measured in degrees always include the unit symbol, for example 45°.

Common radian measurements can be taken from breaking up the unit circle.

Converting between degrees and radians

\(1 \text{ radian} = \dfrac{180^\circ}{\pi}\) and \(1^\circ = \dfrac{\pi}{180}\)

Therefore to convert from Degrees to Radians

\[x^{\circ} \times \frac{\pi}{180}\]

and to convert from Radians to Degrees

\[x \times \frac{180^{\circ}}{\pi}\]

Have a look at the worked example before trying some for yourself.

Worked Example

Convert 45° to radians

\[x^{\circ} \times \frac{\pi}{180}\]

\[\begin{align}&= 45^{\circ} \times \frac{\pi}{180} \\ &= \frac{45 \times \pi}{180} \\ &= \frac{\pi}{4}\end{align}\]


Worked Example

Convert \(5\pi\) to degrees

\[x \times \frac{180^{\circ}}{\pi}\]

\[\begin{align}&= 5\pi \times \frac{180^{\circ}}{\pi} \\ &= \frac{5\pi \times 180^{\circ}}{\pi} \\ &= 900^{\circ}\end{align}\]


Defining Circular Functions: Sine, Cosine and Tangent

Circular functions can be defined using a point \(P\) on a unit circle.

Unit circle on cartesian plane with coordinates labelled. A line from origin (O) to point P is drawn, followed by a dashed line from P to the point on the x-axis labelled ‘A’, creating a right-angled triangle OAP. The internal angle of this triangle, measured counterclockwise from the x axis, is theta.

Point \(P\) of the right-angled triangle \(\triangle AOP\) is located on the unit circle. We can describe the position of \(P\) using \((x,y)\) coordinates.  The value of \(x\) and \(y\) will depend on the angle \(\theta\). We can measure \(\theta\) as the angle from the \(x-\)axis to the line \(\overline{OP}\).

Cosine and Sine

We can use trigonometry to determine the coordinates of  \(P(x, y)\).

From the right-angled triangle \(\triangle OAP\) we can see that:

\(\overline{OA} = x \)

\(\overline{OP} = 1\); as for the unit circle \(r=1\)

And using the principles of trigonometry  we can then define:

\(\begin{align}\cos\left(\theta\right) &= \dfrac{Adjacent}{Hypotenuse} \\ \cos\left(\theta\right) &=\dfrac{x}{1} \\ \cos\left(\theta\right) &= x\end{align}\)

Therefore, the \(x\)-coordinate of \(P\) is defined as \(x = \cos(\theta);  \theta \in \mathbb{R}\).

Also, from the right-angled triangle \(\triangle OAP\):

\(\overline{AP} = y \)

\(\overline{OP} =1\); as it is the unit circle \(r=1\)

And from trigonometry:

\(\begin{align}\sin\left(\theta\right) &= \dfrac{Opposite}{Hypotenuse} \\ \sin\left(\theta\right) &= \dfrac{y}{1} \\ \sin\left(\theta\right) &= y\end{align}\)

Therefore, the \(y\)-coordinate of \(P\) is defined as \(y = \sin(\theta); \theta \in \mathbb{R}\).

Point \(P\) can then be defined in terms of  \(\sin\) and \(\cos\):

Upper right quadrant of a unit circle on cartesian plane with coordinates labelled. A line from origin (O) to point P is drawn, followed by a dashed line from P to the point on the x-axis labelled ‘A’, creating a right-angled triangle OAP. The internal angle of this triangle. measured off x axis, is theta. Coordinates of point P are (cos(theta), sin(theta))

The value of the \(x\)- and \(y\)-coordinates therefore varies only with the value of \(\theta\).

Importantly, we could also describe the same point \(P\), with the same coordinates \((\cos\theta, \sin\theta)\) using many other angles for \(\theta\), depending on how we measure \(\theta\).

Unit circle on cartesian plane with coordinates labelled. A line from origin (O) to point P is drawn, followed by a dashed line from P to the point on the x-axis labelled ‘A’, creating a right-angled triangle OAP. Coordinates of point P are (cos(theta), sin(theta)). The internal angle of this triangle, measured counterclockwise from the x axis to line OP, is theta_1. Theta_2 is measured as a full rotation around the origin from the the \(x\)-axis, and then to line OP, creating an angle that is equal to theta_1+2pi. Theta_3 is two turns around the origin and then to line OP, and is equal to theta_2 +2pi.

We can measure the angle \(\theta_1\), directly from the \(x\)-axis to \(\overline{OP}\),  or we can measure a full rotation around the origin from the the \(x\)-axis, and then to \(\overline{OP}\) as \(\theta_2\).  Similarly, we could measure twice around the origin and then to \(\overline{OP}\) as \(\theta_3\), and could continue like this to infinity. The angle of a full rotation around a unit circle is \(2\pi\).  Adding or subtracting integer multiples of \(2\pi\) still results in the same coordinates, \((\cos\theta, \sin\theta)\), meaning the relationship between \(P\) and \(\theta\) is many-to-one, and results in a periodicity that is fundamental to the behaviour of sine and cosine function.

Tangent

The last function to examine is \(\tan(\theta)\).
By drawing a tangent line at the point \(x = 1\) and extending the \(OP\) to intersect the tangent at \(C\), the coordinates of \(C\) are \((1,y)\) and two similar triangles have been created.

A graph of the unit circle, with a right-angled triangle inscribed. The hypotenuse extends from the origin O(0,0) to a point P on the circle in the first quadrant. Dashed lines indicate the triangle's legs, which correspond to the x- and y-coordinates of the point on the circle. The diagram represents a geometric interpretation of trigonometric functions, with the horizontal leg representing cosine and the vertical leg representing sine of the angle formed with the positive x-axis. Additionally, there is a dashed tangent line to the unit circle at point B(1, 0), corresponding to x=1. This extends to point C(1, y). A second right angled triangle is formed between this tangent, point P and point C. The length of the indicated tangent is tan(theta)

\(\triangle OAP\) and \(\triangle OBC\) are similar triangles. They have the same internal angle \(\theta\) and an equal ratio between their sides. That is

\(\overline{BC}:\overline{OB}=\overline{AP}:\overline{OA}\)

Taking the lengths of these lines from theunit circle, we can also write this as

\(y:1=\sin(\theta):\cos(\theta)\)

These equivalent ratios then imply

\(\begin{align} y &= \frac{\sin(\theta)}{\cos(\theta)} \\ y &= \tan(\theta)\end{align}\)

As \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\) then when \(\cos(\theta) = 0, \tan(\theta)\) is undefined as you can’t divide anything by 0. This means the domain of the function for \(\tan(\theta)\) is \(\theta \in \mathbb{R}\setminus \left\{\theta: \cos(\theta) = 0\right\}\)

This is read as the values at which \(\tan(\theta)\) exists include all possible values of \(\theta\) except for values where \(\cos(\theta) = 0\).

As \(\cos(\theta) = 0\) at odd multiples of \(\frac{\pi}{2}\), then \(\tan(\theta)\) is undefined when

\(\theta = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \pm\frac{5\pi}{2},\ \text{etc.}\)

Symmetry Properties of the Unit Circle

The symmetry of the unit circle can be observed through dividing the circle into four quadrants. This symmetry will be valid for all possible values of \(\theta\).

We can use the symmetry of the unit circle to illustrate if \(\cos, \sin\) and \(\tan\) are positive or negative in the remaining quadrants. Click on the hotspots.

Exact Values of Circular functions for Common Angles

While a calculator could provide all values of \(\sin(\theta)\), \(\cos(\theta)\) or \(\tan(\theta)\),  some values within the unit circle can be easily calculated with exact values by understanding some special triangles.

Sin, Cos and Tan of 0 and \(\dfrac{\pi}{2}\left(90^{\circ}\right)\)

The unit circle shows that the coordinates of \((x,y)\) when \(\theta = 0\) is \((1,0)\), and \((0, 1)\) when \(\theta=\frac{\pi}{2}\).

Unit circle on cartesian plane of radius 1. An angle of 0 and 2 has been drawn out showing the relationship between these angles and the x and y intercepts with coordinates (0, 1) and (1,0)

We know that for any point, \((x,y)\), on the unit circle \(x=\cos(\theta)\) and \(y = \sin(\theta)\). Therefore, we can determine exact values when \(\theta=0\).

\(\begin{align}\ \cos(0) &= 1 \\ \sin(0) &=0\end{align}\)

As \(\tan(\theta) = \dfrac{sin(\theta)}{\cos(\theta)}\) then

\(\tan(0) = \dfrac{0}{1} = 0\)

When \(\theta = \frac{\pi}{2}, (x,y)\) is \(\left(0,1\right)\)

Therefore,

\(\begin{align} \cos\left(\frac{\pi}{2}\right) &= 0 \\ \sin\left(\frac{\pi}{2}\right) &= 1\end{align}\)

As \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), then

\(\tan(\frac{\pi}{2}) = \frac{1}{0} =\text{undefined}\)

Sin, Cos and Tan of \(\frac{\pi}{6}\left(30^{\circ}\right)\) and \(\frac{\pi}{3}\left(60^{\circ}\right)\)

An equilateral triangle of side length 2 has equal angles of 60°.  It can be shown that the a right angle triangle of side lengths \(1, \sqrt{3}\) and \(2\) will have angles  60° and  30°. Therefore, the exact values of sin, cos and tan for \(\dfrac{\pi}{6}\left(30^{\circ}\right)\) and \(\dfrac{\pi}{3}\left(60^{\circ}\right)\) can be determined from this triangle.

Left: An equilateral triangle of side length 2 and a dashed line from the apex to the base. Right: a new triangle formed by half of the left hand triangle with side lengths 2 (hyp), 1(base) and 3 with angles 60°(pi/3) and 30°(pi/6).

Sin, Cos and Tan of \(\frac{\pi}{4} \left(45^{\circ}\right)\)

A right angled isosceles triangle has two equal angles of 45°. By taking the two equal side lengths as 1, and by Pythagoras' theorem the hypotenuse is \(\sqrt{2}\). The exact values for sin, cos and tan of \(\frac{\pi}{4}\left(45^{\circ}\right)\) can be determined from this triangle.

An isosceles triangle of side lengths 1 and sqrt(2) with equal angles 45°

From the special triangles, we can compile a useful table of exact values for \(\sin, \cos\) and \(\tan\) of common angles.

\(\theta\)\(\dfrac{\pi}{6}\quad(30^{\circ})\)\(\dfrac{\pi}{3}\quad(60^{\circ})\)\(\dfrac{\pi}{4}\quad(45^{\circ})\)
\(\cos(\theta)\) \(\dfrac{\sqrt{3}}{2}\) \(\dfrac{\sqrt{1}}{2}\) \(\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}\)
\(\sin(\theta)\) \(\dfrac{\sqrt{1}}{2}\) \(\dfrac{\sqrt{3}}{2}\) \(\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}\)
\(\tan(\theta)\) \(\dfrac{\sqrt{3}}{2}\) \(\sqrt{3}\) \(1\)

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Worked Example

Use the special triangles tables to find the exact value of \(\sin\left(\frac{\pi}{6}\right), \cos\left(\frac{\pi}{6}\right), \tan\left(\frac{\pi}{6}\right)\)

  1. \(\dfrac{\pi}{6} = 30^{\circ}\)
  2. From the special triangles exact values table:

    3. \[\begin{align} \sin\left(\frac{\pi}{6}\right) &= \frac{1}{2} \\ \cos\left(\frac{\pi}{6}\right) &= \frac{\sqrt{3}}{2} \\ \tan\left(\frac{\pi}{6}\right) &= \frac{\sqrt{3}}{3} \end{align}\]

  3. From these values we can also plot a point on the unit circle that corresponds to a radius line at angle \(\frac{\pi}{6}\) from the horizontal

Unit circle drawn with angle of pi/6 and coordinates (sqrt(3)/2,1/2)

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Worked Example

Using symmetry properties of the unit circle, calculate the following:

  1. \(\sin\left(\dfrac{5\pi}{6}\right)\)

    2. Unit circle on a cartesian plane with a line with angle of 5pi/6 = pi-pi/6

    By symmetry properties, \(\frac{5\pi}{6}\) is the same angle as \(\frac{\pi}{6}\), but measured clockwise from the horizontal \(\pi-\frac{\pi}{6}\).

    We know the exact value of \(\sin\left(\frac{\pi}{6}\right)\) from the special triangles table above,  and we know that solutions of \(\sin(\theta)\) are positive in the second quadrant.

    Therefore,

    \(\sin\left(\frac{5\pi}{6}\right) = \sin\left(\pi - \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\)

  2. \(\cos\left(\frac{7\pi}{6}\right)\)

    3. By symmetry properties

    \(\cos\left(\frac{7\pi}{6}\right) = \cos\left(\pi + \frac{\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}\)

    Unit circle on a cartesian plane with a line with angle of 7pi/6 = pi+pi/6

  3. \(\tan\left(\dfrac{11\pi}{6}\right)\)

    4. By symmetry properties

    \(\tan\left(\frac{11\pi}{6}\right) = \tan\left(2\pi - \frac{\pi}{6}\right) = -\tan\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{3}\)

    Unit circle on a cartesian plane with a line with angle of 11pi/6 = 2pi-pi/6

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The Pythagorean Identity

Trigonometric identities are a result of formulas that will be true for all values of \(\theta\) (or any variable used in its place). One of the most fundamental of these is the Pythagorean identity, which relates the squares of the sine and cosine of an angle to the value of one.

Pythagorean Identity

Consider a point \(P(\theta)\)  in the first quadrant of the unit circle.

Unit circle on a Cartesian plane with a right-angled triangle inscribed in the first quadrant, of side lengths labelled cos(theta), sin(theta) and 1. Point P on the unit circle has coordinates (cos(theta), sin(theta))

From the Pythagoras Theorem:

\[\left(\cos\left(\theta\right)\right)^{2} + \left(\sin\left(\theta\right)\right)^{2} = 1\]

Rewriting \(\left(\cos\left(\theta\right)\right)^{2}\text{ as }\cos^{2}\left(\theta\right)\text{ and }\left(\sin\left(\theta\right)\right)^{2} \text{ as }\sin^{2}\left(\theta\right)\)

The Pythagorean identity is defined as

\[\cos^{2}\left(\theta\right) + \sin^{2}\left(\theta\right) = 1\]

For all values of \(\theta\)

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