The Cosecant, Secant, and Cotangent Ratios
In trigonometry, the cosecant, secant, and cotangent functions are
the reciprocals of the three primary trigonometric ratios:
sine, cosine, and tangent.
These relationships are defined as:
\[
\csc \theta = \frac{1}{\sin \theta}, \qquad
\sec \theta = \frac{1}{\cos \theta}, \qquad
\cot \theta = \frac{1}{\tan \theta}.
\]
1. Relationship between Sine and Cosecant
The graph of \(y = \csc \theta\) is derived from \(y = \sin \theta\) by taking
the reciprocal of each sine value.
Wherever \(\sin \theta = 0\), the cosecant function is undefined, resulting
in vertical asymptotes at those points.
2. Relationship between Cosine and Secant
The secant function is the reciprocal of the cosine function:
\[
\sec \theta = \frac{1}{\cos \theta}.
\]
Like the cosecant graph, the secant graph has vertical asymptotes
where \(\cos \theta = 0\).
The curves of \(\sec \theta\) occur where \(\cos \theta\) reaches its maximum and minimum values.
3. Relationship between Tangent and Cotangent
The cotangent function is the reciprocal of tangent:
\[
\cot \theta = \frac{1}{\tan \theta}.
\]
It is undefined where \(\tan \theta = 0\), producing vertical asymptotes at those points.
The graph of \(\cot \theta\) decreases from \(+\infty\) to \(-\infty\) between its asymptotes.
4. Important Observations
- \(\sin \theta\) and \(\csc \theta\) have the same period of \(360^\circ\) (or \(2\pi\) radians).
- \(\cos \theta\) and \(\sec \theta\) also share the same period \(360^\circ\) (or \(2\pi\) radians).
- \(\tan \theta\) and \(\cot \theta\) both have period \(180^\circ\) (or \(\pi\) radians).
- The reciprocal graphs have asymptotes where their corresponding base functions are zero.
5. Reciprocal Identities Summary
\[
\sin \theta = \frac{1}{\csc \theta}, \qquad
\cos \theta = \frac{1}{\sec \theta}, \qquad
\tan \theta = \frac{1}{\cot \theta}.
\]