The Unit Circle and Trigonometric Ratios

Recall that a circle centred at the origin with radius \(1\) is called a unit circle and has equation \(x^2 + y^2 = 1\).

Any point on both the unit circle and the terminal arm of some standard position angle \(\theta\) can be written \(P\,(\cos(\theta), \sin(\theta))\).

Two observations can be made at this point.

We know that \(\tan(\theta) = \dfrac{y}{x}\), and, on the unit circle, \(x = \cos(\theta)\) and \(y = \sin(\theta)\), so \(\tan(\theta) = \dfrac{\sin(\theta)}{\cos(\theta)}\).

This is true for all values of \(\theta\) in the domain of \(\tan(\theta)\), and is called a quotient identity.

Also, since every point is on the unit circle \(x^2 + y^2 = 1\), we can substitute \(x = \cos(\theta)\) and \(y = \sin(\theta)\).

Therefore, \((\cos(\theta))^2 + (\sin(\theta))^2 = 1\). This is usually written \(\cos^2 (\theta) + \sin^2 (\theta) = 1\).

This relationship is true for all values of \(\theta\) and is called the Pythagorean identity.

We will do more with identities later in this unit.

We have also defined three reciprocal trigonometric ratios:

\(\csc(\theta) = \dfrac{1}{\sin(\theta)} \qquad \sec(\theta) = \dfrac{1}{\cos(\theta)} \qquad \cot(\theta) = \dfrac{1}{\tan(\theta)}\)

Given any point \(P\,(x,y)\) on the terminal arm of a standard position angle, we can determine the value of any or all of the six trigonometric ratios.

If that point is also on the unit circle, the process of determining the value of any or all of the six trigonometric ratios is easier.

Point \(P\, \left( -\dfrac{3}{5} , -\dfrac{4}{5} \right)\) is on the terminal arm of standard position angle \(\theta\). Determine the exact values of each of the six trigonometric ratios.

Solution

A sketch is generally helpful.

First, we calculate the value of \(r\).

\(r^2 = x^2 + y^2 = \left( -\dfrac{3}{5} \right)^2 + \left( -\dfrac{4}{5} \right)^2 = \dfrac{9}{25} + \dfrac{16}{25} = 1\)

and \(r = 1\) since \(r \gt 0\). Since \(r = 1\), the point \(P\,\left( -\dfrac{3}{5}, -\dfrac{4}{5} \right)\) is on the unit circle and \((x,y) = (\cos(\theta), \sin(\theta))\). Then,