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Math Help - General solutions

  1. #1
    Member roshanhero's Avatar
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    General solutions

    I am really good at finding general solutions of the trigonometric equation by looking at the graph and by using CAST rule,but when I looked at some books there were some formulas of sin,cos and tan which will give all the solutions of the given equation which I couldnot understand.Please help me through this with the formulas to find all the solutions of sin,cos and tan.
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  2. #2
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    Quote Originally Posted by roshanhero View Post
    I am really good at finding general solutions of the trigonometric equation by looking at the graph and by using CAST rule,but when I looked at some books there were some formulas of sin,cos and tan which will give all the solutions of the given equation which I couldnot understand.Please help me through this with the formulas to find all the solutions of sin,cos and tan.
    You are speaking in general and we cannot see what you mean.
    How about posting an example equation or question or whatever that will give us someting to play or work on?
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  3. #3
    Member roshanhero's Avatar
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    Allright let's have an equation tan theta=1
    We can have the general solution of theta as 45 degrees and 180+45=225 degrees.But I want to find other solutions of theta by using formula,so is there any specific formula of finding solutions of tan,sin and cos.[/COLOR]
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by roshanhero View Post
    Allright let's have an equation tan theta=1
    We can have the general solution of theta as 45 degrees and 180+45=225 degrees.But I want to find other solutions of theta by using formula,so is there any specific formula of finding solutions of tan,sin and cos.[/COLOR]
    If \tan\vartheta =1, then \vartheta=\tan^{-1}1=\frac{\pi}{4}

    Since there is no restriction on \vartheta, we can attempt to generalize this.

    Since \tan\vartheta>0 when 0\leq\vartheta\leq\frac{\pi}{2} and \pi\leq\vartheta\leq\frac{3\pi}{2}, we can see that we can have a \pi radian difference between the angles [either postive or negative].

    So we can have \dots,~-\frac{7\pi}{4},~-\frac{3\pi}{4},~\frac{\pi}{4},~\frac{5\pi}{4},~\fr  ac{9\pi}{4},~\dots

    This can be generalized as \vartheta=\frac{(4n+1)}{4}\pi;~n\in\mathbb{Z}

    Does this process make sense?

    --Chris
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    Quote Originally Posted by roshanhero View Post
    Allright let's have an equation tan theta=1
    We can have the general solution of theta as 45 degrees and 180+45=225 degrees.But I want to find other solutions of theta by using formula,so is there any specific formula of finding solutions of tan,sin and cos.[/COLOR]
    Umm, let me see if I understand now what you mean.

    You want to find the sin, cos, tan, cot, sec and csc of theta from just the given
    tan(theta) = 1.

    You want it done by formulas. Not by the reference right triangle of theta.

    Okay.

    Then we will go to the Pythagorean trig identity because from there we can establish relationships between two trig functions in one equation.

    sin^2(theta) +cos^2(theta) = 1 -----the Pythagorean trig identity.

    Since the given equation, tan(theta) = 1, in in tan only, then we will use the Pythagorean trig identy where tan is involved.

    Divide both sides of the said identity by cos^2(theta),
    tan^2(theta) +1 = sec^2(theta)
    Substitution,
    1^2 +1 = sec^2(theta)
    sec^2(theta) = 2
    sec(theta) = sqrt(2) ----------answer.

    cos(theta) = 1/sec(theta) = 1/sqrt(2) -------answer.

    tan(theta) = sin(theta) / cos(theta)
    sin(theta) = tan(theta) *cos(theta) = 1 * 1/sqrt(2) = 1/sqrt(2) ------answer.

    csc(theta) = 1/sin(theta) = 1 / (1/sqrt(2)) = sqrt(2) -----answer.

    cot(theta) = 1/tan(theta) = 1/1 = 1 --------answer.

    Those are all for theta being in the first quadrant only. If you want to solve for the different trig functions values when theta is in the 3rd quadrant also, then just be carefull with th + and - signs.
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  6. #6
    Member roshanhero's Avatar
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    Thanks for the help specially for the generalised solutions of tan.But is there other formulas of generalised solutions of sin and cos as well.
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  7. #7
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    Quote Originally Posted by roshanhero View Post
    Thanks for the help specially for the generalised solutions of tan.But is there other formulas of generalised solutions of sin and cos as well.
    Just use the same procedure for the tan.
    Go to the basic Pythagorean trig identity, etc, etc...
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