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Math Help - Why do trigonometric functions have reciprocals?

  1. #1
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    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?
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  2. #2
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    \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
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  3. #3
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    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.
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  4. #4
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    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?
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  5. #5
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    Quote Originally Posted by Slappydappy View Post
    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?
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  6. #6
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    Quote Originally Posted by Slappydappy View Post
    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
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  7. #7
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    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.
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