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Math Help - Help with equation of angle x

  1. #1
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    Help with equation of angle x

    I'm trying to understand how to solv x.

    Would be glad if someone could give me a hand.
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  2. #2
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    Hello Reno
    Quote Originally Posted by Reno View Post
    I'm trying to understand how to solv x.

    Would be glad if someone could give me a hand.
    Look at the diagram I've attached, where I've divided the angles x and A into two parts each.

    Here's what you need to find x:
    P = \arctan\left(\frac{b}{a}\right)

    Q = \arctan\left(\frac{a}{b}\right)

    S = A - Q

    \frac{c}{\sin S}= \frac{\sqrt{a^2+b^2}}{\sin(180^o- (R+S))}

    x = P+R
    Can you put all this together?

    Grandad
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  3. #3
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    Thanks alot. But I must say that my skills has been better, I would be happy if you could help me further.
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  4. #4
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    Hello Reno

    First, a rather belated Welcom to Math Help Forum!
    Quote Originally Posted by Reno View Post
    Thanks alot. But I must say that my skills has been better, I would be happy if you could help me further.
    I'm assuming that we have to find an expression for x in terms of a, b and A. It will be pretty complicated if we put it all together. I think it's probably best left in terms of P,Q,R and S.

    So, as I said in my previous post:
    \frac{c}{\sin S}=\frac{\sqrt{a^2+b^2}}{\sin(180^o-(R+S))}
    (Do you understand how I got this? I used the Sine Rule on the lower triangle, having first used Pythagoras' Theorem on the upper triangle.)

    So, noting that \sin\theta = \sin(180^o-\theta):
    \frac{c}{\sin S}=\frac{\sqrt{a^2+b^2}}{\sin(R+S)}

    \Rightarrow \sin(R+S) = \frac{(\sqrt{a^2+b^2})\sin S}{c}

    \Rightarrow R = \arcsin\left(\frac{(\sqrt{a^2+b^2})\sin S}{c}\right)-S
    Then I should leave the answer as:
    x = P+R, where
    P = \arctan\left(\frac{b}{a}\right)

    R = \arcsin\left(\frac{(\sqrt{a^2+b^2})\sin S}{c}\right)-S

    S = A-Q

    Q = \arctan\left(\frac ab\right) (or you could simply say Q = 90^o-P.)
    But you could combine these into a single expression by substituting for each of the variables P,Q,R,S if you wish.

    Grandad
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  5. #5
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    <br />
S = A - (PI + Q)<br />

    Thanks for the excellent help. Now I know my Sin_rule.
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