# Thread: Trigonometry Problems [involves inverse functions]

1. ## Trigonometry Problems [involves inverse functions]

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.... =)

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

$\displaystyle \sin[2\arctan{x} + \arccos(2x)]$

for the second expression ...

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

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

using the sum identity for cosine ...

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

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

sub these values into the sum expression above ...

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

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

4. Originally Posted by FallenStar117
So how is it you found the value of arctan(3)?
I didn't ... I said let $\displaystyle \theta = \arctan(3)$ , which also says $\displaystyle \tan{\theta} = 3$ ... I used this fact to calculate $\displaystyle \sin{\theta}$ and $\displaystyle \cos{\theta}$ which was necessary evaluate the original expression.

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