Reciprocal Trigonometric Ratios and Their Pythagorean Identities
Reciprocal Trigonometric Functions
The reciprocal trigonometric functions are:
- Cosecant (csc): Reciprocal of sine $$\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}$$
- Secant (sec): Reciprocal of cosine $$\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$$
- Cotangent (cot): Reciprocal of tangent $$\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} = \frac{\text{adjacent}}{\text{oppposite}}$$
When working with reciprocal functions, remember that there are asymptotes. They're undefined when their denominators equal zero!
Common MistakeMany students assume that $\sec x$ is the reciprocal of $\sin x$ and $\csc x$ is the reciprocal of $\cos x$. It's actually the other way round from what you'd expect, so $\sec x = \frac{1}{\cos x}$ and $\csc x = \frac{1}{\sin x}$.
Pythagorean Identities for reciprocal trigonometric functions
Just like before, we can start with the Pythagorean theorem for a right-angled triangle:
$$o^2 + a^2 = h^2$$
Before, we divided both sides by $h^2$. However, we can also divide both sides by $o^2$ or $a^2$, and see what comes up.
Dividing both sides by $o^2$:
$$1 + \frac{a^2}{o^2} = \frac{h^2}{o^2}$$
But $\cot\theta$ = \frac{a}o$, and $\csc\theta = \frac{h}o$. So, we can say:
$$1 + \cot^2\theta = \csc^2\theta$$
Returning to the original Pythagorean theorem, we can also divide both sides by $a^2$ to get:
$$\frac{o^2}{a^2} + 1 = \frac{h^2}{a^2}$$
But $\tan\theta = \frac{o}a$, and $\sec\theta = \frac{h}a$. So:
$$\tan^2\theta + 1 = \sec^2\theta$$
Thus, to summarize the three Pythagorean trigonometric identities:
- $$\sin^2 \theta + \cos^2 \theta = 1$$
- $$1+ \cot^2\theta = \csc^2 \theta$$
- $$\tan^2 \theta + 1 = \sec^2 \theta$$
Graphing
The graphs of $y = \sec x$ and $y = \csc x$ look similar:
Both have range $y\in(-\infty, -1]\cup[1, \infty)$, and both have the characteristic "alternating buckets" shape.
However, $\sec x$ has asymptotes at $x = \frac{2k+1}2\pi, k\in\mathbb{Z}$ (i.e. odd multiples of $\frac{\pi}2$, or $\frac{\pi}2, \frac{3\pi}2, \frac{5\pi}2$, etc.) $\csc x$ has asymptotes at $x = k\pi, k\in\mathbb{Z}$ (i.e. multiples of $\pi$).
$\sec x$ is an even function and has a y-intercept at (0, 1), while $\csc x$ is an odd function and has an asymptote at $x = 0$ (so no y-intercept exists).
This makes sense if you think of them as reciprocals of the normal trigonometric functions. $\csc x$ has asymptotes where $\sin x$ has zeroes, and $\sec x$ has asymptotes where $\cos x$ has zeroes.
The graph of $y = \cot x$ looks like:
