tangent of inverse trignometric function

**viet** · Feb 4th 2007, 12:54 PM

a) express in term of X :
$\sin\left( 2 \tan^{-1}(x)\right)$

b) Find: $\tan\left( \sin^{-1}(\frac {5}{8}) + \cos^{-1}(\frac {1}{6})\right)$

**Soroban** · Feb 4th 2007, 01:48 PM

Hello, viet!

a) Express in term of $x$ : . $\sin\left(2\tan^{-1}(x)\right)$

We will need the identity: . $\sin2\theta \:=\:2\sin\theta\cos\theta$

Recall that an inverse trig value is an angle.

Let $\theta = \tan^{-1}(x)\quad\Rightarrow\quad\tan\theta = x$

Since $\tan\theta = \frac{x}{1} = \frac{opp}{adj}$ , we have: $opp = x,\:adj = 1$
. . and Pythagorus gives us: $hyp = \sqrt{x^2+1}$
Hence: . $\sin\theta = \frac{x}{\sqrt{x^2+1}},\;\cos\theta = \frac{1}{\sqrt{x^2+1}}$

The problem becomes: . $\sin(2\theta) \:=\:2\sin\theta\cos\theta$

. . . . . . $=\:2\left(\frac{x}{\sqrt{x^2+1}}\right)\left(\frac {1}{\sqrt{x^2+1}}\right) \;=\;\boxed{\frac{2x}{x^2+1}}$

b) Find: . $\tan\left[\sin^{-1}\left(\frac {5}{8}\right) + \cos^{-1}\left(\frac {1}{6}\right)\right]$

We have: . $\tan\left[\underbrace{\sin^{-1}\left(\frac{5}{8}\right)}_{\alpha} + \underbrace{\cos^{-1}\left(\frac{1}{6}\right)}_{\beta}\right]$

Then: . $\sin\alpha = \frac{5}{8}\quad\Rightarrow\quad opp = 5,\:hyp = 8\quad\Rightarrow\quad adj = \sqrt{39}$
. . Hence: . $\tan\alpha = \frac{5}{\sqrt{39}}$

And: . $\cos\beta = \frac{1}{6}\quad\Rightarrow\quad adj = 1,\:hyp = 6\quad\Rightarrow\quad opp = \sqrt{35}$
. . Hence: . $\tan\beta = \frac{\sqrt{35}}{1} = \sqrt{35}$

The problem becomes: . $\tan(\alpha + \beta)\;=\;\frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta} \;=\;$ $\frac{\frac{5}{\sqrt{39}} + \sqrt{35}}{1 - \frac{5}{\sqrt{39}}\cdot\sqrt{35}}$

Multiply top and bottom by $\boxed{\sqrt{39}\!:\;\;\frac{5 + \sqrt{1365}}{\sqrt{39} - 5\sqrt{35}}}$

I hope I didn't make any stupid blunders . . .

**thedoge** · Feb 4th 2007, 07:23 PM

I used your method for part B on a similar problem and it is not working.

numbers were 5/6 for the sin, 5/9 for the cos.

ended with (((5/sqrt(11))+(sqrt(56)/5))/(1-(5/sqrt(11))*(sqrt(56)/5)))*sqrt(11)
but it is wrong

**Soroban** · Feb 4th 2007, 08:41 PM

Hello, thedoge!

I used your method for part B on a similar problem and it is not working.

Numbers were 5/6 for the sin, 5/9 for the cos.

Ended with: (((5/sqrt(11))+(sqrt(56)/5))/(1-(5/sqrt(11))*(sqrt(56)/5)))*sqrt(11) .??

but it is wrong

Evidently, your preliminary work is correct.
. . $\tan\alpha = \frac{5}{\sqrt{11}},\;\;\tan\beta = \frac{\sqrt{56}}{5}$

Identity: . $\tan(\alpha + \beta) \;=\:\frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\cdot\tan\beta}$

We have: . $\tan(\alpha + \beta) \;=\;\frac{\frac{5}{\sqrt{11}} + \frac{\sqrt{56}}{5}} {1 - \left(\frac{5}{\sqrt{11}}\right)\!\left(\frac{\sqr t{56}}{5}\right)}$

Multiply top and bottom by $5\sqrt{11}:\;\;\frac{25 + \sqrt{56}\sqrt{11}}{5\sqrt{11} - 5\sqrt{56}}$

**thedoge** · Feb 4th 2007, 08:46 PM

Ah ok. Thank you

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