SOLUTION: Find the exact value of each of the following. Enter an exact answer; decimal approximations will be marked incorrect. tan(arctan(41/15)) arctan(tan(70&#960;/56))

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Question 966873: Find the exact value of each of the following. Enter an exact answer; decimal approximations will be marked incorrect.
tan(arctan(41/15))
arctan(tan(70π/56))

Found 2 solutions by Fombitz, Theo:
Answer by Fombitz(32388) (Show Source):
You can put this solution on YOUR website!
arctan and tan are inverse functions.
tan(arctan(41/15))=41/15
arctan(tan(70π/56))=70π/56

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
by definition, tan(a) = b if and only if arctan(b) = a

therefore:

if tan(a) = b, then arctan(b) = a

tan(arctan(b)) is therefore equal to tan(a) which is equal to b.

this gets you tan(arctan(b)) = b.

if you let b = 41/15, then you get:

tan(arctan(b)) = b becomes tan(arctan(41/15)) = 41/15.

that takes care of your first statement.

now to move on to your second statement.

by definition, arctan(c) = d if and only if tan(d) = c.

therefore, if arctan(c) = d, then tan(d) = c.

arctan(tan(d) is therefore equal to arctan(c) which is equal to d.

this gets you arctan(tan(d)) = d.

if you let d = 70pi/56, then you get:

arctan(tan(70pi/56)) = 70pi/56.

you can confirm these properties are correct by using your calculator.
set your calculator to degree mode.

tan(45) = 1
arctan(1) = 45

this leads to:

tan(arctan(1)) = 1
arctan(tan(45) = 45

SOLUTION: Find the exact value of each of the following. Enter an exact answer; decimal approximations will be marked incorrect. tan(arctan(41/15)) arctan(tan(70&#960;/56))

SOLUTION: Find the exact value of each of the following. Enter an exact answer; decimal approximations will be marked incorrect. tan(arctan(41/15)) arctan(tan(70π/56))