Trigonometry Problems [involves inverse functions]

**FallenStar117** · March 20th 2011, 08:56 AM

So I am having a bit of trouble studying for my trig test coming up....

for the section on inverse trig functions I am a bit confused with some questions

ex: sin[29arctanx)+arcos(2x)]

ex: cos[arctan3 - arcsin(-0.5)]

I understand the basics such as remembering the domains of the different functions and what not... but these questions seem to stump me.

Can you maybe solve one but with a thorough description.... =)

**skeeter** · March 20th 2011, 09:40 AM

I assume the first expression has a typo, and should be

$\sin[2\arctan{x} + \arccos(2x)]<br />$

please confirm.

for the second expression ...

$\cos[\arctan(3) - \arcsin(-0.5)]$

let $\theta = \arctan(3)$ , and you should already know that $\arcsin(-0.5) = -\dfrac{\pi}{6}$

using the sum identity for cosine ...

$\cos\left(\theta + \dfrac{\pi}{6}\right) = \cos{\theta} \cdot \cos\left(\dfrac{\pi}{6}\right) - \sin{\theta} \cdot \sin\left(\dfrac{\pi}{6}\right)$

since $\tan{\theta} = 3$ , then $\cos{\theta} = \dfrac{1}{\sqrt{10}}$ and $\sin{\theta} = \dfrac{3}{\sqrt{10}}$

sub these values into the sum expression above ...

$\cos\left(\theta + \dfrac{\pi}{6}\right) = \dfrac{1}{\sqrt{10}} \cdot \dfrac{\sqrt{3}}{2} - \dfrac{3}{\sqrt{10}} \cdot \dfrac{1}{2} = \dfrac{\sqrt{3} - 3}{2\sqrt{10}}$

**FallenStar117** · March 20th 2011, 05:38 PM

So how is it you found the value of arctan(3)?

**skeeter** · March 20th 2011, 05:55 PM

Originally Posted by FallenStar117

So how is it you found the value of arctan(3)?

I didn't ... I said let $\theta = \arctan(3)$ , which also says $\tan{\theta} = 3$ ... I used this fact to calculate $\sin{\theta}$ and $\cos{\theta}$ which was necessary evaluate the original expression.

**FallenStar117** · March 21st 2011, 08:28 AM

So how would I start solving the first question?
Is the idea the same as the second question...

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