Reciprocal Trigonometric Ratios and Their Pythagorean Identities

Reciprocal Trigonometric Functions

The reciprocal trigonometric functions are:

1. Cosecant (csc): Reciprocal of sine $$\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}$$
2. Secant (sec): Reciprocal of cosine $$\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$$
3. Cotangent (cot): Reciprocal of tangent $$\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} = \frac{\text{adjacent}}{\text{oppposite}}$$

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:

1. $$\sin^2 \theta + \cos^2 \theta = 1$$
2. $$1+ \cot^2\theta = \csc^2 \theta$$
3. $$\tan^2 \theta + 1 = \sec^2 \theta$$

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:

$y = \cot x$ looks like $y = \tan x$, just flipped and shifted by $\frac{\pi}2$. This is not a coincidence – $\cot x = \tan(\frac{\pi}2 - x)$.

$\cot x$ has asymptotes at $x = k\pi$ and zeroes at $\x = \frac{2k+1}2\pi, k\in\mathbb{Z}$. It's also an odd function.

Inverse Circular Functions

Inverse trigonometric functions (also called arcfunctions) are the inverse functions of the trigonometric functions. If the original trigonometric functions turn angles into ratios, inverse trigonometric functions turn ratios back into angles.

The main inverse trigonometric functions are:

1. $\arcsin x$ or $\sin^{-1} x$
2. $\arccos x$ or $\cos^{-1} x$
3. $\arctan x$ or $\tan^{-1} x$

Domains and Ranges:

• $\arcsin x$: Domain $x\in[-1,1]$, Range $y\in[-\frac{\pi}{2}, \frac{\pi}{2}]$
• $\arccos x$: Domain $x\in[-1,1]$, Range $y\in[0, \pi]$
• $\arctan x$: Domain $x\in\mathbb{R}$, Range $y\in(-\frac{\pi}{2}, \frac{\pi}{2})$

Properties

1. Composition with original function:
   • $\sin(\arcsin x) = x$ for $x \in [-1,1]$
   • $\arcsin(\sin x) = x$ for $x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$
2. Similar properties hold for cosine and tangent:
   • $\cos(\arccos x) = x$ for $x \in [-1,1]$
   • $\tan(\arctan x) = x$ for all real $x$

Graphing

The graphs of $y = \arcsin x$ and $y = \arccos x$, as you would expect by now, look very similar:

These are just the original trig functions truncated and reflected over the line $y = x$, as for any inverse function. You can clearly see the restricted domains and ranges.

The graph of $y = \arctan x$ looks like:

The graph just looks like one period of $\tan x$, reflected across the line $y = x$.

This has a domain of all real numbers, which is a reflection of the unrestricted range of $\tan x$. However, $\arctan x$ has horizontal asymptotes at $y = \frac\pi2$ and $y=-\frac\pi2$, which reflects the vertical asymptotes of $\tan x$.