# Learning $\arcsin, \arccos, \arctan$ - how to?

I have a very basic understanding of $$\arcsin, \arccos, \arctan$$ functions. I do know how their graph looks like and not much more beyond that.

Calculate:

Which specific keywords should I google to learn how to solve the following tasks? I think those aren't equations (googling 'cyclometric equations' was a dead end). Perhaps you would like to share with some link to a beginner-friendly learning source?

Thank you.

• inverse trigonometric functions is what I would call them – Henry Lee Dec 5 '18 at 1:06
• also a lot of them just require the compound angle formula or some rearrangment to make them a lot easier – Henry Lee Dec 5 '18 at 1:07
• "cyclometric" is the old name for the inverse trig functions. Any basic book should have sections on how to evaluate these. – Mortified Through Math Dec 5 '18 at 1:11
• these are more simple but a start: tutorial.math.lamar.edu/Extras/AlgebraTrigReview/… – Henry Lee Dec 5 '18 at 1:11
• if you ask specifically about 1 of those questions that you struggle with people will also help – Henry Lee Dec 5 '18 at 1:12

Some good reference as summary from Wikipedia are

and also

Some exercises given requires only to calculate the value for the functions at some point, other else are more tricky and you need to acquire a deeply understanding of the matter.

Refer also to the related

• That should help, thanks. – weno Dec 5 '18 at 1:14
• Do not hesitate to ask for any exercise in particular! Bye – user Dec 5 '18 at 1:15
• I would, but unless I made any significant progress myself, I don't think anyone will solve that for me. :) – weno Dec 5 '18 at 1:17
• That’s nice but in case of doubt I would happy to check that with you. Bye – user Dec 5 '18 at 1:19

I would say there are three things you are expected to do on this list. One is to know the trig functions of special angles, so for 4 you should know that $$\tan \frac \pi 4=1,$$ so $$\arctan 1=\frac \pi 4$$ Watch out for the ranges specified for the inverse trig functions. Second is that $$\sin(\arcsin (x))=x$$. When you have $$\arcsin (\sin(x))$$ you may be shifted by factors of $$\pi$$. Finally when you have $$\sin(\arccos(\frac 13))$$ draw a right triangle with $$\cos$$ of one angle $$\frac 13$$, so it is a $$1-\sqrt 8-3$$ triangle and find the sine of the angle, here $$\frac 13\sqrt 8$$.

• Hey, thanks. I understood everything except for: "When you have arcsin(sin(x)) you may be shifted by factors of π". Would you give a little bit more insight on this? – weno Dec 5 '18 at 1:19
• The range of $\arcsin$ is defined as $[-\frac \pi 2,\frac \pi 2]$, so if we are asked for $\arcsin(\sin(\frac {9\pi}4))$ the answer is $\frac \pi 4,$ not $\frac {9 \pi}4$ – Ross Millikan Dec 5 '18 at 1:22
• Those indeed are the tricky questions which I was referring to! – user Dec 5 '18 at 1:23
• I don't know which post should I accept, since you both helped me a lot. I'll do a random.org roll. – weno Dec 5 '18 at 1:29

This lecture has been particularly helpful on understanding the subject: