Unit Circle

A unit circle from the name itself defines a circle of unit radius. In the following guide, you will learn more about the unit circle.

Unit Circle

The location of a point which is a distance of one unit from a fixed point is called a unit circle.

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Step by step guide to the unit circle

A unit circle is a circle with a radius of \(1\) unit. The unit circle is generally shown on the Cartesian coordinate plane. The unit circle is represented algebraically using the second-degree equation with two variables \(x\) and \(y\). The unit circle is used in trigonometry and is useful for finding the values of the trigonometric ratios of sine, cosine, and tangent.

Equation of a unit circle

The general equation of a circle is \((x-a)^2+(y-b)^2=r^2\), which represents a circle having the center \((a, b)\) and the radius \(r\). This equation of a circle is simplified to show the equation of a circle. A unit circle is formed with its center at the point\((0, 0)\), which is the origin of the coordinate axes and a radius of \(1\) unit. So the equation of the unit circle is:

\(\color{blue}{x^2+y^2=1}\)

Note: the above equation satisfies all the points lying on the circle across the four quadrants.

Finding trigonometric functions using a unit circle

Consider a right triangle located in a unit circle on the Cartesian coordinate plane. The radius of the circle represents the hypotenuse of the right triangle. The radius vector makes an angle \(θ\) with the positive \(x\)-axis and the coordinates of the endpoint of the radius vector are \((x, y)\). Here the values of \(x\) and \(y\) are the lengths of the base and the altitude of the right triangle. By applying this to trigonometry, we can find the values of the trigonometric ratio as follows:

\(\color{blue}{sin\:\theta=\frac{Altitude}{Hypoteuse}=\frac{y}{1}}\)

\(\color{blue}{cos\:\theta=\frac{Base}{Hypotenuse}=\frac{x}{1}}\)

Note: now we have \(sin\:\theta = y\), \(cos\:\theta = x\), and using this, we have \(tan\:\theta = \frac{y}{x}\).

Unit circle chart in radians

The unit circle shows a complete angle of \(2π\) radians. And the unit circle is divided into four quadrants at angles of \(\frac{\pi }{2}\), \(\pi\), \(\frac{3\pi }{2}\), \(2\pi\) respectively. More in the first quadrant at the angles of \(0\), \(\frac{\pi }{6}\), \(\frac{\pi }{4}\), \(\frac{\pi }{3}\), \(\frac{\pi }{2}\) are the standard values, which are applicable to the trigonometric ratios.

The points on the unit circle for these angles represent the standard angle values of the cosine and sine ratio. By looking closely at the figure below, the values are repeated in four quadrants, but with a change in the sign.

Unit circle and trigonometric identities

For a right triangle placed in a unit circle in the cartesian coordinate plane, with hypotenuse, base, and altitude measuring \(1\), \(x\), \(y\) units respectively, the unit circle identities can be given as:

  • \(\color{blue}{sin\:\theta =\frac{y}{1}}\)
  • \(\color{blue}{cos\:\theta =\frac{x}{1}}\)
  • \(\color{blue}{tan\:\theta =\frac{sin\:\theta }{cos\:\theta }=\frac{y}{x}}\)
  • \(\color{blue}{sec\:\left(\theta \right)=\frac{1}{x}}\)
  • \(\color{blue}{csc\:\left(\theta \right)=\frac{1}{y}}\)
  • \(\color{blue}{cot\:\left(\theta \right)=\frac{cos\:\theta }{sin\:\theta }=\frac{x}{y}}\)

Unit circle pythagorean identities

The three Pythagorean identities in trigonometry are as follows:

  • \(\color{blue}{sin^2\theta +cos^2\theta =1}\)
  • \(\color{blue}{1+tan^2\theta =sec^2\theta}\)
  • \(\color{blue}{1+cot^2\theta =cosec^2\theta}\)

Unit circle table

The unit circle table is used to list the coordinates of points on a unit circle that corresponds to common angles using trigonometric ratios:

Unit circle in complex plane

A unit circle is composed of all complex numbers of absolute value as \(1\). Therefore, it has the equation of \(|z| = 1\). Any complex number \(z = x + iy\) will lie on the unit circle with equation given as \(x^2+y^2=1\).

A unit circle can be thought of as unit complex numbers in a complex plane, for example, a set of complex numbers \(z\) given by the form,

\(\color{blue}{z=e^{it}= cos\:t+ i\:sin\:t=cis\:\left(t\right)}\)

Unit Circle – Example 1:

Does point \(A (\frac{1}{2}, \frac{1}{2})\) lie on the unit circle?

Solution:

The equation of a unit circle is: \(x^2+y^2=1\)

Substituting \(x =\frac{1}{2}\) and \(y = \frac{1}{2}\), we get:

\(=\frac{1}{2}^2+\frac{1}{2}^2\)

\(=\frac{1}{4}+\frac{1}{4}\)

\(=\frac{2}{4}=\frac{1}{2}\)

\(≠ 1\)

Therefore \((\frac{1}{2}, \frac{1}{2})\) doesn’t lie on the unit circle.

Exercises for Unit Circle

Find the value of a function using a unit circle chart.

  1. \(\color{blue}{sin\:900^{^{\circ }}}\)
  2. \(\color{blue}{cos\:240^{^{\circ }}}\)
  3. \(\color{blue}{tan\:225^{^{\circ }}}\)
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{0}\)
  2. \(\color{blue}{-\frac{1}{2}}\)
  3. \(\color{blue}{1}\)

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