Finding Possible Values Of Trigonometric Expression

**ArcherSam** · March 31st 2012, 05:28 PM

@MHF

How is the answer to this trigonometric expression?

$\begin{array}{l}\cos x = \frac{1}{{11}}\\\frac{{\sec x - \tan x}}{{\sin x}} = \pm \frac{{121\sqrt {30} - 660}}{{60}}\end{array}$

Even when I start off I get the wrong answer for tangent x.

$\begin{array}{l}\tan x = \pm \sqrt {{{\sec }^2}x - 1} \\\tan x = \pm \sqrt {{{(11)}^2} - 1} \\\tan x = \pm 20\sqrt 3 - 1 \end{array}$

Sam

**Prove It** · March 31st 2012, 05:43 PM

Originally Posted by ArcherSam

@MHF

How is the answer to this trigonometric expression?

$\begin{array}{l}\cos x = \frac{1}{{11}}\\\frac{{\sec x - \tan x}}{{\sin x}} = \pm \frac{{121\sqrt {30} - 660}}{{60}}\end{array}$

Even when I start off I get the wrong answer for tangent x.

$\begin{array}{l}\tan x = \pm \sqrt {{{\sec }^2}x - 1} \\\tan x = \pm \sqrt {{{(11)}^2} - 1} \\\tan x = \pm 20\sqrt 3 - 1 \end{array}$

Sam

I think you'll find that $\displaystyle \begin{align*} \pm \sqrt{11^2 - 1} = \pm \sqrt{121 - 1} = \pm \sqrt{120} = \pm 2\sqrt{30} \end{align*}$ ...

**ArcherSam** · March 31st 2012, 06:00 PM

@Prove It

Lol, I should have checked if $\pm 20\sqrt 3$ was equal to $\sqrt {120}$ .

With this I get:

$\begin{array}{l} \cos x = \frac{1}{{11}}\\ \sec x = \frac{1}{{\frac{1}{{11}}}} = 11\\ \tan x = \pm \sqrt {{{(11)}^2} - 1} = \sqrt {121 - 1} = \sqrt {120} = \pm 2\sqrt {30} \\ \sin x = \pm \sqrt {1 - {{(\frac{1}{{11}})}^2}} = \sqrt {1 - \frac{1}{{121}}} = \sqrt {\frac{{121}}{{121}} - \frac{1}{{121}}} = \sqrt {\frac{{120}}{{121}}} = \pm \frac{{2\sqrt {30} }}{{11}} \end{array}$

Then I am stuck!

**Prove It** · March 31st 2012, 06:21 PM

So now you need to evaluate $\displaystyle \begin{align*} \frac{\sec{x} - \tan{x}}{\sin{x}} \end{align*}$ ...

**ArcherSam** · March 31st 2012, 06:56 PM

@Prove It

It was that easy!

$\begin{array}{l} \frac{{11 - ( + 2\sqrt {30} )}}{{ + \frac{{2\sqrt {30} }}{{11}}}}\\ 11(\frac{{11 - (2\sqrt {30} }}{{2\sqrt {30} }})\\ \sqrt {30} (\frac{{121 - 22\sqrt {30} }}{{2\sqrt {30} }})\\ \frac{{121\sqrt {30} - 22(30)}}{{60}}\\ \frac{{121\sqrt {30} - 660}}{{60}} \end{array}$

Thanks!

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