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

1. ## 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!

2. ## 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*}.

3. ## Re: Value of Sin 2x Given (Equation w/ Several Trig Functions)

Originally Posted by Lexielai
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

$\displaystyle r = \sin x+\cos x + 1$

$\displaystyle t = \sin 2x = 2 \sin x \cos x$

The given equation can then be written as

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

It is also easy to show that

$\displaystyle r^2 = 2r+t$