How to solve an equation

**TOZE** · December 21st 2011, 02:55 AM

The equation is:

cotg(X) = (sqrt(3)*cos(50º) + sin(50º))*(1-2*sqrt(3)*cos(50º)+2*sin(50º)) /
(sqrt(3)*sin(50º) - cos(50º))

Using a calculator, I find X = 50º.

How to develop the equation to find the solution?

Thanks.

**HallsofIvy** · December 21st 2011, 03:30 AM

What do you mean by "develop" the equation? The right side is simply a number. Find that number and take its inverse cotangent.

**TOZE** · December 21st 2011, 06:28 AM

Doing what you propose with the calculator we arrive to 50 degrees.
So, if X = 50, it would be possible to "develop" or simplify the trigonometric expression on the second member of the equation to arrive to cotg(50).
Don't you think so, considering that in the expression we have only trigonometric functions of the same angle?

**FernandoRevilla** · December 21st 2011, 09:59 AM

Originally Posted by TOZE

Don't you think so, considering that in the expression we have only trigonometric functions of the same angle?

Consider any angle $\alpha$ instead of $50^0$ and expand $\cot \alpha$ in the following way:

$\cot \alpha =\cot \;[30^0+(\alpha -30^0)]=\frac{\cot 30^0\cdot \cot (\alpha -30^0)-1}{\cot 30^0+\cot (\alpha -30^0)}=$

$\frac{\sqrt{3}\cdot \dfrac{\cos (\alpha-30^0)}{\sin (\alpha-30^0)}-1}{\sqrt{3}+\dfrac{\cos (\alpha-30^0)}{\sin (\alpha-30^0)}}}=\ldots$

Let's see if you get the right side of the equation (with $\alpha$ instead of $50^0$ ).

P.D. I haven't checked it, it is only a proposal.

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