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camicat

Member

Registered: 2020-02-27

Posts: 11

Inverse Trig Functions

Hello,
I am working on Inverse Trig Functions in school and I am honestly super frustrated with it. My teacher keeps giving me videos to watch, but it's really hard for me to focus on videos. I would greatly appreciate it if someone could write out the steps on how to solve these problems so that I can understand the process of it. Even just walking through a practice example for each type of question would be super helpful. Here are some examples of the different types of questions I am working on:

2. Find the principal value of each of the following in radians:

Arccsc (-2)

A. pi/2
B. -pi/6
C. pi/5
D. -pi
E. 5pi/6
F. -2pi

6. Find the principal value of each of the following to the nearest minute:

Arccsc (1.607)

A. 38°64'
B. 39°24'
C. 38°48'
D. 38°32'
E. 39°37'
F. 38°29'

11. Solve for the angle x.

2cos(x) - 2 = -1

A. no solution
B. pi/3 + 2pin
C. pi/3 + 2pin, 5pi/3 + 2pin
D. 5pi/3 + 2pin
E. pi/3
F. pi/3, 5pi/3

Thanks in advance.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Inverse Trig Functions

hi camicat

2. Arccsc (-2)

I would unpick this a bit at a time.  cosec is 1/sine so you are trying to find the angle that makes cosec = -2 which is the angle that makes  sine = 1/-2

So when is sine -0.5 ?

Sine 30 degrees is + 0.5.  To get minus 0.5 you need to be in the third or fourth quadrant.  The principal values for sine are 1st or 4th quadrant so I need -0.5 in the 4th.

That is 360 - 30.

Now into radians.

180 degrees is pi rads so 360-30 is 2pi minus pi/6= 11pi/6

Whoops I don't see that.  Arhh! Yes I do because -pi/6 is the same angle (11/6 anticlockwise brings you to the same place as 1/6 clockwise. Check definition of principal values .... -pi/2 to + pi/2.  Fair enough, -pi/6 it is.

6. So work out 1/1.607  and inverse sine it.  Convert to degrees with x (180/pi)

11. 2cos(x) - 2 = -1

rearrange so you have cos(x) = a number.

There's certainly a solution but it looks like the general solutions are also given.  eg If 80 degrees is a solution (it isn't this is just illustrative) then so is 80 + 360 and 80 + 360 + 360 and 80 + 360 + 360 + 360 etc etc.

You can show them all in one neat expression with 80 + 360n where n is any integer.

So get an answer. Convert to radians. List extra solutions by adding multiples of 2 pi.

But before you choose your answer, also consider what other angle (apart from 80) could also be a solution.  For cosine an answer in the first quadrant also has an answer in the fourth (-80 or 280)  So look closely for a second set of general solutions and then pick the answer that has them all covered.

Hope that helps.  I'll check your answers if you want to post them.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: Inverse Trig Functions

hi camicat

2. Arccsc (-2)

I would unpick this a bit at a time.  cosec is 1/sine so you are trying to find the angle that makes cosec = -2 which is the angle that makes  sine = 1/-2

So when is sine -0.5 ?

Sine 30 degrees is + 0.5.  To get minus 0.5 you need to be in the third or fourth quadrant.  The principal values for sine are 1st or 4th quadrant so I need -0.5 in the 4th.

That is 360 - 30.

Now into radians.

180 degrees is pi rads so 360-30 is 2pi minus pi/6= 11pi/6

Whoops I don't see that.  Arhh! Yes I do because -pi/6 is the same angle (11/6 anticlockwise brings you to the same place as 1/6 clockwise. Check definition of principal values .... -pi/2 to + pi/2.  Fair enough, -pi/6 it is.

6. So work out 1/1.607  and inverse sine it.  Convert to degrees with x (180/pi)

11. 2cos(x) - 2 = -1

rearrange so you have cos(x) = a number.

There's certainly a solution but it looks like the general solutions are also given.  eg If 80 degrees is a solution (it isn't this is just illustrative) then so is 80 + 360 and 80 + 360 + 360 and 80 + 360 + 360 + 360 etc etc.

You can show them all in one neat expression with 80 + 360n where n is any integer.

So get an answer. Convert to radians. List extra solutions by adding multiples of 2 pi.

But before you choose your answer, also consider what other angle (apart from 80) could also be a solution.  For cosine an answer in the first quadrant also has an answer in the fourth (-80 or 280)  So look closely for a second set of general solutions and then pick the answer that has them all covered.

Hope that helps.  I'll check your answers if you want to post them.

Bob

Nice reply. Great work. I am looking forward to our math journey here.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,424

Re: Inverse Trig Functions

See the link : Inverse Trigonometric Ratios.

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: Inverse Trig Functions

See the link : Inverse Trigonometric Ratios.

What about taking the derivative of inverse trig functions? How is this done?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Inverse Trig Functions

eg. 

Differentiate wrt x:

   

This last step (to get the answer as a function of x) uses

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: Inverse Trig Functions

eg. 

Differentiate wrt x:

   

This last step (to get the answer as a function of x) uses

Bob

The idea is to use implicit differentiation.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Inverse Trig Functions

Well, yes; but it's a two step process.  First change the equation to one that you know how to differentiate, then implicit is forced on you.

B

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathland

Member

Registered: 2021-03-25

Posts: 444

Re: Inverse Trig Functions

Well, yes; but it's a two step process.  First change the equation to one that you know how to differentiate, then implicit is forced on you.

B

Thank you. [comment removed by Admin]  I do use your hints and steps to solve the posted questions on paper. [comment removed by Admin}