Trigonometric Ratios

For \(P\,(x,y)\), any point on the terminal arm of some standard position angle \(\theta\), with \(r\) the distance from the origin to \(P\), the definitions for the sine, cosine and tangent ratios are given in terms of \(x\), \(y\), \(r\), and \(\theta\):

These definitions were covered in earlier courses.

If this same point \(P~(x,y)\) also lies on the unit circle, since \(r = 1\), then the definitions become

It follows that \(P\,(x,y)\) can be written \(P\, \Big( \cos (\theta), \sin (\theta) \Big)\).

That is, for any point on both the unit circle and the terminal arm of some standard position angle \(\theta\),

• the \(x\)-coordinate is \(\cos (\theta)\),
• the \(y\)-coordinate is \(\sin (\theta)\), and
• the \(y\)-coordinate divided by the \(x\)-coordinate is \(\tan (\theta)\).

Reciprocal Trigonometric Ratios

The ratios sine, cosine, and tangent are referred to as the primary trigonometric ratios.

A second set of ratios exist that are called the reciprocal trigonometric ratios.

The reciprocal trigonometric ratios are cosecant, (csc); secant, (sec); and cotangent, (cot).

The reciprocal trigonometric ratios are defined as follows:

On the unit circle, since \(r = 1\), the definitions become

\(\csc (\theta) = \dfrac{1}{y} \qquad \sec (\theta) = \dfrac{1}{x} \qquad \cot (\theta) = \dfrac{x}{y}\)

That is, for any point on both the unit circle and the terminal arm of some standard position angle \(\theta\), the reciprocal of the \(x\)-coordinate is \(\sec (\theta)\), the reciprocal of the \(y\)-coordinate is \(\csc (\theta)\) and \(\cot (\theta)\) is the \(x\)-coordinate divided by the \(y\)-coordinate.

The point \(P\,(-6,3)\) is on the terminal arm of an angle \(\theta\) in standard position where \(0 \leq \theta \leq 2 \pi\). Determine the exact values of the six trigonometric ratios.

Solution

A sketch is generally helpful in visualizing the problem.

We know that \(x^2 + y^2 = r^2\) with \(x = -6\) and \(y = 3\).

Substituting, \(r^2 = (-6)^2 + (3)^2 = 45\) and \(r = \sqrt{45} = 3\sqrt{5}\) follows since \(r \gt 0\).

We can now calculate the six trigonometric ratios:

The terminal arm of a standard position angle \(\theta\) intersects the unit circle at point \(Q\), a point in the third quadrant with \(x\)-coordinate \(-\dfrac{1}{2}\). Determine the exact values of \(\cos (\theta)\) and \(\cot (\theta)\).

Solution

Again, a sketch is generally helpful in visualizing the problem.

Let \(y\) represent the \(y\)-coordinate of point \(Q\).

Since \(Q\) is in the third quadrant, \(y \lt 0\).

For any point \((x,y)\) on the unit circle, we know that \(x^2 + y^2 = 1\).

Substituting \(x = -\dfrac{1}{2}\), we get \(\left( -{\dfrac{1}{2}} \right)^2 + y^2 = 1\).

This simplifies to \(\dfrac{1}{4} + y^2 = 1\) and it follows that \(y^2 = \dfrac{3}{4}\).

Therefore, \(y = \pm \dfrac{\sqrt{3}}{2}\) but \(y \lt 0\) in quadrant \(3\), so \(y = \dfrac{\sqrt{3}}{2}\) is inadmissible and it follows that \(y = -\dfrac{\sqrt{3}}{2}\).

On the unit circle, \(\cos (\theta) = x = -\dfrac{1}{2}\) and \(\cot (\theta )= \dfrac{x}{y} = -\dfrac{1}{2} \div \left( -\dfrac{\sqrt{3}}{2} \right) = \dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}\).

In this module, we

• defined and sketched standard position angles,
• defined coterminal angles and reference angles,
• developed expressions for general coterminal angles and found coterminal angles in a specified domain,
• developed an equation for a circle with centre \((0,0)\) and radius \(r\),
• introduced the unit circle,
• reviewed the definitions of the primary trigonometric functions,
• introduced reciprocal trigonometric functions with definitions, and
• applied the definitions to the unit circle.