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Thread: Trigonometric functions and three consecutive numbers or non consecutive number

  1. #1
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    Post Trigonometric functions and three consecutive numbers or non consecutive number

    When the triangle is constant (meaning that the base equals to one),the sum of the angles in degrees are 180

    Can you mix dimensionless function or angles with lenght in triangles?For example two sides are composed of a distance of $0.85+0.4=1.25$ and at the same time $0.4=\cos\theta$and the base is $1$?

    For consecutive number or non consecutive numbers $x<y<z$ I have the following example:

    $(((\frac{\sqrt\frac{y}{z}}{(1-\frac{x}{z})\times\sqrt\frac{x+z}{z-x}})\times\frac{x}{z})+\sqrt\frac{z-y}{z})\times((1-\frac{x}{z})\times\sqrt\frac{(x+z)}{(z-x)})=\sin A$

    $(\frac{\sqrt\frac{y}{z}}{(1-\frac{x}{z})\times\sqrt\frac{x+z}{z-x}})-(((\frac{\sqrt\frac{y}{z}}{(1-\frac{x}{z})\times\sqrt\frac{x+z}{z-x}})\times\frac{x}{z})+\sqrt\frac{z-y}{z})\times(\frac{x}{z})=\cos A$

    $\sqrt\frac{(z-y)}{z}=\cos B$

    $\sqrt\frac{y}{z}=\sin B$

    $\frac{x}{z}=\cos C$

    $((1-\frac{x}{z})\times\sqrt\frac{(z+x)}{(z-x)})=\sin C$

    $(\sqrt{\frac{y}{z}}\times\frac{x}{z})+\sqrt\frac{ z-y}{z}\times((1-\frac{x}{z})\times\sqrt\frac{(z+x)}{(z-x)})=\sin A$

    $(-\sqrt{\frac{z-y}{z}})\times\frac{x}{z}+\sqrt{\frac{y}{z}}\times( (1-\frac{x}{z})\times\sqrt\frac{(z+x)}{(z-x)})=\cos A$

    The following variables $a,b,c$ represent the length of the sides of the triangles.The angles have no unit but the lengths do.

    $\frac{\sin A}{\sin C}=a$

    $\frac{\sin B}{\sin C}=b$

    $\frac{\sin C}{\sin C}=c$

    h=altitude $\frac{h_c}{h_a}=a$

    $\frac{h_c}{h_b}=b$

    $\frac{h_c}{h_c}=c$

    $((((\frac{\sin B}{\sin C})\times\cos C)+\cos B)\times\sin C)=\sin A$

    $(\frac{\sin B}{\sin C})-((((\frac{\sin B}{\sin C})\times\cos C)+\cos B)\times\cos C)=\cos A$

    $((((\frac{\sin A}{\sin C})\times\cos C)+\cos A)\times\sin C)=\sin B$

    $(\frac{\sin A}{\sin C})-((((\frac{\sin A}{\sin C})\times\cos C)+\cos A)\times\cos C)=\cos B$

    And you can obtain all six theta of trigonometric functions with consecutive numbers or non consecutive where $x<y<z$.
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  2. #2
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    Re: Trigonometric functions and three consecutive numbers or non consecutive number

    What? I have no idea what you are trying to do. Here is my confusion:
    1. The interior angles of triangles always sum to 180 degrees.
    2. What are "consecutive" vs "non-consecutive" numbers in your understanding? Your example uses decimals, which are rarely consecutive. Consecutive numbers imply a sequence of numbers. While it is true that rational numbers are countably infinity, there is no single canonical mapping from counting numbers to rational numbers that would allow one to define "consecutive" numbers canonically.
    3. Angles are not dimensionless. They do not lack units. Their unit is degrees or radians.

    Anyway, I think what you are looking for is the Law of Sines and the Law of Cosines. The Law of Sines states that for any triangle with sides $a,b,c$ and angles opposite to those sides $A,B,C$ respectively:
    Trigonometric functions and three consecutive numbers or non consecutive number-triangle.png
    the following equalities hold:

    $\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}$
    $c^2 = a^2+b^2-2ab\cos C$
    $b^2 = a^2+c^2-2ac\cos B$
    $a^2 = b^2+c^2-2bc\cos A$

    This is true for any triangle, regardless of "consecutive" or "non-consecutive" numbers or any other conditions you may want to place on the triangle.
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  3. #3
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    Re: Trigonometric functions and three consecutive numbers or non consecutive number

    About consecutive number and non consecutive:
    For example 17. 21. 25. are consecutive and 17. 24. 27 are not (depends of the sequence on).Both of them equal to variables x<y<z
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  4. #4
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    Re: Trigonometric functions and three consecutive numbers or non consecutive number

    Quote Originally Posted by Larrousse View Post
    About consecutive number and non consecutive:
    For example 17. 21. 25. are consecutive and 17. 24. 27 are not (depends of the sequence on).Both of them equal to variables x<y<z
    Then you are looking for where the numbers are sequential in an arithmetic progression or non-sequential in any arithmetic progression. Regardless, the Law of Sines and Law of Cosines still applies.
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