Cosecant, Secant and Cotangent Ratios

The reciprocals of the sine, cosine and tangent ratios are defined as

\[\textbf{secant} \theta = \sec \theta = \frac{1}{\cos \theta}\] \[\textbf{cosecant} \theta = \textbf{cosec} \theta = \frac{1}{\sin \theta}\] \[\textbf{cotangent} \theta = \cot \theta = \frac{1}{\tan \theta}\]

Tip: Look at the third letter to determine the reciprocal of the trigonometric ratio.

Please note that:

\(\cos^n \theta = (\cos \theta)^n\), \(\sin^n \theta = (\sin \theta)^n\), \(\tan^n \theta = (\tan \theta)^n\) for \(n \geq 1\),

i.e., \(\cos^2 \theta = (\cos \theta)^2\), \(\sin^3 \theta = (\sin \theta)^3\), \(\tan^5 \theta = (\tan \theta)^5\)

But \(\cos^{-1} \theta \neq (\cos \theta)^{-1}\), \(\sin^{-1} \theta \neq (\sin \theta)^{-1}\) and \(\tan^{-1} \theta \neq (\tan \theta)^{-1}\)

\(y=\cos^{-1}x\iff\cos y=x\text{ and}\hspace{0.5em}0\leq y\leq\pi\), for \(-1\leq x\leq1\).

\(y=\sin^{-1}x\iff\sin y=x\text{ and}\hspace{0.5em}-\frac{\pi}{2}\leq y\leq\frac{\pi}{2}\), for \(-1\leq x\leq1\).

\(y=\tan^{-1}x\iff\tan y=x\text{ and}\hspace{0.5em}-\frac{\pi}{2}<y<\frac{\pi}{2}\), for real \(x\).

The six trigonometric ratios

For any point \(P(x, y)\) on the rotating arm, with \(r=\sqrt{x^2+y^2}\) always positive, the six ratios are defined below. When \(\theta\) is acute this matches SOH-CAH-TOA: \(y\) is the opposite side, \(x\) the adjacent side and \(r\) the hypotenuse.

Each quotient requires a non-zero denominator. In particular, \(\tan\theta\) and \(\sec\theta\) require \(\cos\theta\ne0\); \(\mathrm{cosec}\theta\) and \(\cot\theta=\cos\theta/\sin\theta\) require \(\sin\theta\ne0\). The expression \(1/\tan\theta\) agrees with \(\cot\theta\) only where both are defined. At \(\cos\theta=0\), cotangent is zero although tangent is undefined.