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Thread: Trigonometric Proof

  1. #1 Trigonometric Proof 
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    I wrote a proof yesterday for the trigonometric identity that states that . Is my proof correct?

    Here it is:

    1. and

    2. . since , where .

    3. is equal to , which is equal to , which is equal to

    4. .


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  3. #2  
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    Proof by derivation. Looks OK.


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    Is r supposed to be the hypothenuse of a right-angled triangle? If so, your proof only holds for theta smaller than 90 degrees. As far as I know, but please correct me if I'm wrong, tan(v) is by definition equal to sin(v)/cos(v) and thus it would be unmotivated to prove it.
    What I'm getting is that sin(v), cos(v) and tan(v) are not defined from a right-angled triangle, they're defined from the unit circle. There, sin(v) is the y-value and cos(v) the x-value for the point P such that the line that goes from origo at the angle v intersects the circle in P.
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  5. #4  
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    Quote Originally Posted by thyristor
    Is r supposed to be the hypothenuse of a right-angled triangle? If so, your proof only holds for theta smaller than 90 degrees. As far as I know, but please correct me if I'm wrong, tan(v) is by definition equal to sin(v)/cos(v) and thus it would be unmotivated to prove it.
    What I'm getting is that sin(v), cos(v) and tan(v) are not defined from a right-angled triangle, they're defined from the unit circle. There, sin(v) is the y-value and cos(v) the x-value for the point P such that the line that goes from origo at the angle v intersects the circle in P.
    The provided in the proof does not refer to the hypotenuse of a triangle but the length of the pole made by rotating a ray in the coordinate plane, although the pythagorean theorem is used to find the length of of just as is done in a right triangle. The trigonometric function tangent as the quotient of the sine and cosine functions is an identity, therefore there must be a proof to confirm it. The proof holds true for all angles of , and not simply those where For example, , and .
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  6. #5  
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    Ok, but according to http://en.wikipedia.org/wiki/Trigono...ctions#Tangent (see unit circle definitions), tan(v) is actually defined as sin(v)/cos(v).
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  7. #6  
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    on that page, even;
    Quote Originally Posted by wikipedia
    The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case
    Wise men speak because they have something to say; Fools, because they have to say something.
    -Plato

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    Quote Originally Posted by Arcane_Mathematician
    on that page, even;
    Quote Originally Posted by wikipedia
    The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case
    As I wrote in brackets see "unit circle definition".
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  9. #8  
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    that's the derived definition, yes, but not the explicit definition. The explicit definition is the one listed above it, in the tangent function section that you directly linked. Tangent is defined: and it is derived, in similar manner to the way Ellatha did, that and it was not originally defined that way.
    Wise men speak because they have something to say; Fools, because they have to say something.
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  10. #9  
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    Thrystor,
    What you're referring to are what as known as identities. There are many others such as reciporical, cofunction, and periodicity identities.

    From these we could express as or or even .

    All of these identities are derived from the six fundamental trigonometric functions, however, which are the following:





    The previously mentioned identities are derived from the above definitions, and require proofs to confirm them.
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  11. #10 Re: Trigonometric Proof 
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    Quote Originally Posted by Ellatha
    I wrote a proof yesterday for the trigonometric identity that states that . Is my proof correct?

    Here it is:

    1. and

    2. . since , where .

    3. is equal to , which is equal to , which is equal to

    4. .
    .

    is the definition of . No proof is required.
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  12. #11  
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    Not that Calculus by Howard Anton is the ultimate authority or anything, but the definition given therein for the tan(theta) = y/x, and it's referencing the familiar figure with coordinates, circle, theta, x, y and r.
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  13. #12 Re: Trigonometric Proof 
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by Ellatha
    I wrote a proof yesterday for the trigonometric identity that states that . Is my proof correct?

    Here it is:

    1. and

    2. . since , where .

    3. is equal to , which is equal to , which is equal to

    4. .
    .
    Thanks for the support!

    is the definition of . No proof is required.
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  14. #13  
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    Quote Originally Posted by Ellatha
    Thrystor,
    What you're referring to are what as known as identities. There are many others such as reciporical, cofunction, and periodicity identities.

    From these we could express as or or even .

    All of these identities are derived from the six fundamental trigonometric functions, however, that is .

    The previously mentioned identities are derived from the above definitions, and require proofs to confirm them.
    Ok, but as far as I know, most mathematicians define tan(v) as sin(v)/cos(v).
    If one uses or definition of tangent, indeed it is motivated to prove that tan(v)=sin(v)/cos(v).
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    Quote Originally Posted by Arcane_Mathematician
    that's the derived definition, yes, but not the explicit definition. The explicit definition is the one listed above it, in the tangent function section that you directly linked. Tangent is defined: and it is derived, in similar manner to the way Ellatha did, that and it was not originally defined that way.
    I don't understand what you mean. It says on wikipedia "Above, only sine and cosine were defined directly by the unit circle, but other trigonometric functions can be defined by:
    tan(theta)=sin(theta)/cos(theta)..."
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  16. #15 Tangent Definition 
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    I respectfully disagree that sin over cosine is the definition of tangent.

    Basic Trigonometry defines the tanget of an angle as the length of the side opposite the angle divided by the length of the side adjacent to the angle in a right triangle.
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  17. #16  
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    I agree with paulfr2.
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  18. #17 Re: Tangent Definition 
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    Quote Originally Posted by paulfr2
    I respectfully disagree that sin over cosine is the definition of tangent.

    Basic Trigonometry defines the tanget of an angle as the length of the side opposite the angle divided by the length of the side adjacent to the angle in a right triangle.
    No, basic trigononetry defines the cosine as the projectin of of a line of unit length and angle theta with respect to the x-axis onto the x-axis and the sine asw the projection onto the y-axis the tangent is the ratio of the sine to the cosine.

    There are more advanced definitions based on power series.

    But there is no need to define the tangent as a ratio of sine to cosine as that is the definition.

    You get to your assertion regarding ratios of sides of right triangles through and arguement based on ordinary Eudlicean geometry, but it is it not normally the fundamental definition.
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  19. #18 Re: Tangent Definition 
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by paulfr2
    I respectfully disagree that sin over cosine is the definition of tangent.

    Basic Trigonometry defines the tanget of an angle as the length of the side opposite the angle divided by the length of the side adjacent to the angle in a right triangle.
    No, basic trigononetry defines the cosine as the projectin of of a line of unit length and angle theta with respect to the x-axis onto the x-axis and the sine asw the projection onto the y-axis the tangent is the ratio of the sine to the cosine.

    There are more advanced definitions based on power series.

    But there is no need to define the tangent as a ratio of sine to cosine as that is the definition.

    You get to your assertion regarding ratios of sides of right triangles through and arguement based on ordinary Eudlicean geometry, but it is it not normally the fundamental definition.
    Nope.
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  20. #19 Re: Tangent Definition 
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    Dr Rocket,

    Power series are not definitions, they are approximations.

    If the sine cosine ratio is the definition of tangent, where was this first defined and by whom ?

    And if Euclid is not fundamental, then exactly who is ???
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  21. #20 Re: Tangent Definition 
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    Quote Originally Posted by paulfr2
    Dr Rocket,

    Power series are not definitions, they are approximations.

    If the sine cosine ratio is the definition of tangent, where was this first defined and by whom ?

    And if Euclid is not fundamental, then exactly who is ???
    I'd just like to comment this:

    Since e^(ix)=cos(x)+isin(x), e^(ix)+e^(-ix)=cos(x)+isin(x)+cos(-x)+isin(-x)=2cos(x) which gives that cos(x)=(1/2)*(e^(ix)+e^(-ix)) from which we can define sin(x) by sin(x)=cos(pi/2-x). Since e^(ix)=1+ix-x^2/2!-ix^3/3!+x^4/4! ... we get that
    cos(x)=(1/2)*((1+x-x^2/2!-ix^3/3!+x^4/4!...)+(1-x-x^2/2!+x^3/3!+x^4/4!...))=(1-x^2/2!+x^4/4!-x^6/6!...) which is a power series.
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  22. #21  
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    Thyristor

    Your previous post on page 1 cited this as evidence that tan = sin / cos
    http://en.wikipedia.org/wiki/Trigono...ctions#Tangent

    But it clearly shows the right triangle side ratio as the definition.

    Did you not even read this page before posting it ?
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  23. #22  
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    Quote Originally Posted by paulfr2
    Thyristor

    Your previous post on page 1 cited this as evidence that tan = sin / cos
    http://en.wikipedia.org/wiki/Trigono...ctions#Tangent

    But it clearly shows the right triangle side ratio as the definition.

    Did you not even read this page before posting it ?
    It says: "The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin. The unit circle definition provides little in the way of practical calculation; indeed it relies on right triangles for most angles." and a bit down in the same section "Above, only sine and cosine were defined directly by the unit circle, but other trigonometric functions can be defined by:" tan(theta)=sin(theta)/cos(theta).
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  24. #23  
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    I find Wiki informative but keep in mind that the info is user based. While I have found it to be generally correct it's not an authority.
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  25. #24 Method not Definition 
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    The unit circle definitions are in fact the right triangle definitions.
    It just happens that the hypotenuse has a value of unity [thus unit circle].
    So the ratios reduce to a triangle side length.
    But they are really ratios and the definitions are the same.
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  26. #25 Re: Method not Definition 
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    Quote Originally Posted by paulfr2
    The unit circle definitions are in fact the right triangle definitions.
    It just happens that the hypotenuse has a value of unity [thus unit circle].
    So the ratios reduce to a triangle side length.
    But they are really ratios and the definitions are the same.
    With the unit circle, definitions for right triangels alone are not needed. The unit circle definitions are general and can thus be applied to all angles, while your right triangle definitions are not general and can thus not be applied to all angles
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