Asked by Anonymus
Why is the answer to the exact value of arctan(tan(3pi/4))= -pi/4
Answers
Answered by
Reiny
Since arctan(anything) has multiple answers, the mathematical convention is to use the answer closest to zero
arctan(tan(3π/4))= 3π/4 would be correct, but -π/4 is closer to zero
(let's look at degrees, most people can relate with degrees better than with radians)
3π/4 radians = 135° an angle in quadrant II
so we would have arctan(tan(135°))= -45°
we know that tan 135° = -1
but so is tan-45°, tan 315°, tan -225° or an infinite number of other angles
If we take arctan(-1) our calculator has been programmed to state the closest angle to zero, and we would get -45
test it with other trig inverses such as " arcsin(sin 210°) " , and you should get -30°
I hope this makes sense.
arctan(tan(3π/4))= 3π/4 would be correct, but -π/4 is closer to zero
(let's look at degrees, most people can relate with degrees better than with radians)
3π/4 radians = 135° an angle in quadrant II
so we would have arctan(tan(135°))= -45°
we know that tan 135° = -1
but so is tan-45°, tan 315°, tan -225° or an infinite number of other angles
If we take arctan(-1) our calculator has been programmed to state the closest angle to zero, and we would get -45
test it with other trig inverses such as " arcsin(sin 210°) " , and you should get -30°
I hope this makes sense.
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