Thread: Why do we have the secant and cosecant functions?

As far as I understand, everything can be expressed using the four basic trigonometric functions (sine, cosine, tangent, cotangent), so why were the reciprocals of sine and cosine, the cosecant and secant functions, introduced?
Is it just a matter of simplification and compression in writing, or are there some historical reasons?

I appreciate your input!

P.S. I just can't use 'sec' and 'csc'. I always end up converting them to sine and cosine :P And my results are always expressed in sin and cos, is that wrong?

Originally Posted by mrfour44

As far as I understand, everything can be expressed using the four basic trigonometric functions (sine, cosine, tangent, cotangent), so why were the reciprocals of sine and cosine, the cosecant and secant functions, introduced?
Is it just a matter of simplification and compression in writing, or are there some historical reasons? P.S. I just can't use 'sec' and 'csc'. I always end up converting them to sine and cosine :P And my results are always expressed in sin and cos, is that wrong?

First question: why did you not include cotangent function in this question?

It seems that the answer to that may help answer your question.
After all the secant function has the same relation to the cosine function as the cotangent to the tangent. Etc.

Moreover, in calculus the derivative of is . So it is important.

Sorry, I even seldom use the cotangent function so I didn't actually register it as a reciprocal, though, I do see your point. I'd always expressed the derivative of the tangent function as 1 over cosine squared. :P

Add:
Haven't sine, cosine and tangent always been primary, as in you see them in a triangle? Though, that can be said for any trig function. I think of them in a loving fashion when I think back to my good old high school trigonometry basics :P

You could "get away" with using only sine, cosine, and tangent. Secant, Cosecant, and Cotangent are included for completeness (A right triangle has 3 sides. There are 3(2)= 6 ways of combining two of them in a fraction.) and for simplicity- there are times when it is easier to use sec(x) rather than 1/cos(x).

Unit Circle Definitions




It also helps to know what the words mean:

Sine - Perpendicular.
Cosine - Works in Conjunction with Sine.
Tangent - The length of the Tangent to the circle.
Secant - The length that is cut by the tangent.
Cosecant - Works in Conjunction with the Secant.
Cotangent - Works in Conjunction with the Tangent.

Actually, the meaning of the words is different.

Sine comes from the Latin sinus, -us, n. that means bending/curve/fold/bay. It is actually an incomplete translation of the Sanskrit word jiva (chord). All the other functions have the co- prefix because their values in the complementary angles to the sine's argument were equal to the original sine. You could say that it's in conjunction, but that is not the origin since "conjunctio, -onis, n." carried the meaning of union, mutual love and familiarity or conjunction (grammar).

The tangent and secant functions do derive their names from the nouns tangens, -ntis, n., meaning tangent, and secans, -ntis, n., meaning secant. The tangent touches (tango, tangere), while the secant cuts through (seco, secare).

LOL i just answered my own question. Thank you all :P