How to begin this?

[jtw2e2]

How does one approach this? Are we looking for an angle?

 

[stapel]

The input to the tangent function is an angle measure; it spits out a number.

The input to the inverse tangent function is a number; it spits out an angle measure.

So find the angle measure from within the range of the inverse tangent function, such that the tangent value would be the same as the tangent value for pi/10. (This output can be the same as the input, but you'll want to get into the habit of checking first, before sort-of "cancelling" the inverses.)

:wink:

 

[jtw2e2]

Re:

The input to the tangent function is an angle measure; it spits out a number.

The input to the inverse tangent function is a number; it spits out an angle measure.

So find the angle measure from within the range of the inverse tangent function, such that the tangent value would be the same as the tangent value for pi/10. (This output can be the same as the input, but you'll want to get into the habit of checking first, before sort-of "cancelling" the inverses.)

:wink:

Ok, I see how this output is the same as the input, but what might be an example in which one receives a different output from a sin(arcsinx) formula?

 

[stapel]

\(\displaystyle \sin^{-1}\left(\sin\left(\frac{3pi}{4}\right)\right)\, =\, \frac{\pi}{4}\)

:wink:

 

[chrisr]

In a right angled triangle, Sin(A) gives the ratio of two side lengths,
the inverse Sine of that ratio recovers the angle "A".

For simplicity initially, the below refers to a unit-radius circle,
centered at the origin (0,0).

In a circle (also has angles > 90 degrees), Sine now represents the y co-ordinate
(the right angled triangle is used to calculate it's distance from the origin).

Any angle other than 0, 90, 180 and 270 degrees has a "twin" angle with the
exact same y co-ordinate.
Picture that situation on the unit circle.
Since calculators are programmed to return only one of those two angles (unfortunately)
using the inverse Sine function, there is a second solution not mentioned
for angles other than 0, 90, 180 and 270 degrees.

It's not so much that you get a different answer.
Any y co-ordinate except on the axes can reference two angles
but your calculator will not give you both
(unless there is a make I don't know of).

 

[Deleted member 4993]

Re: Re:

The input to the tangent function is an angle measure; it spits out a number.

The input to the inverse tangent function is a number; it spits out an angle measure.

So find the angle measure from within the range of the inverse tangent function, such that the tangent value would be the same as the tangent value for pi/10. (This output can be the same as the input, but you'll want to get into the habit of checking first, before sort-of "cancelling" the inverses.)

:wink:

Ok, I see how this output is the same as the input, but what might be an example in which one receives a different output from a sin(arcsinx) formula?

function                 derived from                       domain                    range 
Arcsin            inverse of sine function                [?1; +1]                 [??/2; +?/2] 
Arccos            Arccos x = ?/2 ? Arcsin x             [?1; +1]                   [0; ?] 
Arctan              inverse of tangent function           all reals                (??/2; +?/2) 
Arccot              Arccot x = ?/2 ? Arctan x              all reals                (0; ?) 
Arcsec           Arcsec x = Arccos(1/x)             (??; ?1], [1; ?)            [0; ?] 
Arccsc           Arccsc x = Arcsin(1/x)              (??; ?1], [1; ?)            [??/2; +?/2]

reference:

http://oakroadsystems.com/twt/inverse.htm