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Math Help - Value of Sin 2x Given (Equation w/ Several Trig Functions)

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    Value of Sin 2x Given (Equation w/ Several Trig Functions)

    Hey everybody!

    This question really has me scratching my head. I probably forgot whatever formula, rule, or whatever is supposed to help me solve this.

    What is the value of Sin 2x given that sin x + cos x + tan x + cot x + sec x + csc x = 7?

    To be honest, I'm clueless on this one. Any help would be appreciated!
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    Re: Value of Sin 2x Given (Equation w/ Several Trig Functions)

    I would start by trying to turn the difficult equation into something in terms of sin(2x). Converting to sines and cosines and getting a common denominator would be a start. Also remember that $\displaystyle \begin{align*} \sin{(2x)} \equiv 2\sin{(x)}\cos{(x)} \end{align*}$.
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    Re: Value of Sin 2x Given (Equation w/ Several Trig Functions)

    Quote Originally Posted by Lexielai View Post
    Hey everybody!

    This question really has me scratching my head. I probably forgot whatever formula, rule, or whatever is supposed to help me solve this.

    What is the value of Sin 2x given that sin x + cos x + tan x + cot x + sec x + csc x = 7?

    To be honest, I'm clueless on this one. Any help would be appreciated!
    Set

    r = \sin x+\cos x + 1

    t = \sin 2x = 2 \sin x \cos x

    The given equation can then be written as

    r-1 + \frac{2r}{t} = 7

    It is also easy to show that

    r^2 = 2r+t
    Thanks from Lexielai
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