Silly Old Harry Caught A Herring Trawling Off America.

It is also worth knowing that

        sin x
tan x = -----
        cos x

You can visit trigonometric identities for many, many more similar mind-numbingly exciting pieces information. Don't laugh, some of this stuff is actually pretty useful.

Three additional, somewhat redundant trigonometric ratios are found by taking the reciprocals of the first three.


          1     hypotenuse
sec x = ----- = ----------
        cos x    adjacent


          1     hypotenuse
csc x = ----- = ----------
        sin x    opposite


          1      adjacent
cot x = ----- = ----------
        tan x    opposite

It's not recommended to use these for the purposes of trying to rearrange trig formulae. Having both sine, cosine and tangent and their reciprocals in the same equation can be terribly, terribly confusing and lead to errors. Stick wholly with sines and cosines (and tangents if you like, but they can knocked out, see above) until right at the end when you want to make the final expression as small as possible.

The terms secant, cosecant and cotangent for the reciprocals of sine, cosine and tangent are, as far as I can tell, only kept in use i) for completeness, and ii) because they are shorter to say and write under certain circumstances. They allow us to say for instance that

-- tan x dx = sec x tan x

which is an elegant way of expressing a standard definition, as opposed to the relatively long-winded

d               sin x
-- tan x dx = --------
dx             cos2 x
which is something of a mouthful to pronounce. Finally, never confuse the following
(sin x)-1 and sin-1 x

(cos x)-1 and cos-1 x

(tan x)-1 and tan-1 x

On the left, trigonometric functions inverted; on the right, inverse trigonometric functions. NEVER confuse the two. No good will come of this. See also hyperbolic functions.