# Why do trigonometric functions have reciprocals?

• Jan 16th 2011, 04:20 PM
Slappydappy
Why do trigonometric functions have reciprocals?
I've gotten to the point in my trigonometry studies where I am looking at Cosecant and Sine, and I see that Cosecant and Sine are reciprocals of each other. Now this means that multiplied together produce 1 (on the Unit Circle) and I'm assuming everywhere else.

Now why is their relationship a reciprocal one? I see as Sine gets smaller, Cosecant (and Cotangent) get larger. Is there a website that specifically describes why the changing of one causes a change in the other one (outside of the obvious). I drew pictures of them, so I can visually see the change, but I don't understand why the numbers change the way they do.

Does that make sense?
• Jan 16th 2011, 04:30 PM
dwsmith
$\displaystyle\sin{x}=\frac{y}{r} \ \ \ \csc{x}=\frac{1}{\sin{x}}=\frac{1}{\frac{y}{r}}=\f rac{r}{y}$

$\displaystyle\frac{y}{r}\cdot\frac{r}{y}=\frac{yr} {yr}=1$
• Jan 16th 2011, 05:16 PM
DrSteve
Just to express in words what dwsmith is saying: $\csc x$ is usually defined to be the reciprocal of $\sin x$. All of the properties of $\csc x$ follow from this definition.
• Jan 16th 2011, 10:36 PM
Slappydappy
Is there a proof of it somewhere? Like a geometric proof that shows it? I have this image I have been looking at:

http://schools-wikipedia.org/images/226/22690.png

Or am I ahead of myself?
• Jan 17th 2011, 06:44 PM
dwsmith
Quote:

Originally Posted by Slappydappy
Is there a proof of it somewhere? Like a geometric proof that shows it? I have this image I have been looking at:

http://schools-wikipedia.org/images/226/22690.png

Or am I ahead of myself?

What do you want to prove?
• Jan 18th 2011, 05:15 AM
CaptainBlack
Quote:

Originally Posted by Slappydappy
I've gotten to the point in my trigonometry studies where I am looking at Cosecant and Sine, and I see that Cosecant and Sine are reciprocals of each other. Now this means that multiplied together produce 1 (on the Unit Circle) and I'm assuming everywhere else.

Now why is their relationship a reciprocal one? I see as Sine gets smaller, Cosecant (and Cotangent) get larger. Is there a website that specifically describes why the changing of one causes a change in the other one (outside of the obvious). I drew pictures of them, so I can visually see the change, but I don't understand why the numbers change the way they do.

Does that make sense?

The cosec, sec and cot are introduced just to make life more complicated for trig students, they are completely unnecessary

CB
• Jan 18th 2011, 06:11 AM
HallsofIvy
First, there are 3 sides to a right triangle and so 3(2)= 6 ways of "ordering" any two of them: of "abc" the 6 orders of two at at time are "ab", "ac", "ba", "bc", "ca", and "cb". Once you start making fractions of those three sides, since there are two parts (numerator and denominator) there are 6 possible fractions. Calling the "near leg" to an angle $\theta$, "a", the "far leg", b, and the hypotenuse, "c", three of those fractions, $\frac{a}{c}= cos(\theta)$, $\frac{b}{c}= sin(\theta)$ and $\frac{a}{b}= tan(\theta)$, are commonly used.

The other three, $\frac{c}{a}= sec(\theta)$, $\frac{c}{b}= csc(\theta)$, and $\frac{b}{a}= cot(\theta)$ are defined pretty much just to be complete. Although, as much as I like Captain Black's response, there are times when you need to divide by "sine", "cosine", or "cotangent" and then "sec", "csc", or "cot" can be convenient, though they are never necessary.