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

Question

Sorry for asking such question.

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.

Answer 1

Some good reference as summary from Wikipedia are

• List of trigonometric identities

and also

• Inverse trigonometric functions

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

• Reference Request: Book for Trigonometry and Geometry

• Good book on advanced trig

Answer 2

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 $\frac 13\sqrt 8$.

Answer 3

This lecture has been particularly helpful on understanding the subject:

https://academics.utep.edu/Portals/1788/CALCULUS%20MATERIAL/4_7%20INVERSE%20TRIG%20FNS.pdf