How do you evaluate arctan(-1) using a unit circle?

Answer 1

Using a unit circle centered at #(0,0)# in the Cartesian plane
#tan(theta)# is the #y# coordinate value divided by the #x# coordinate value of the intersection of the unit circle and a ray extending from the origin at an angle of #theta#

Asking for #arctan(-1)# is the same as asking to solve
for #theta# in
#tan(theta) = -1#

This will happen when #x# and #y# have equal magnitudes but opposite signs
Within the domain #theta epsilon [0,2pi]#
this only happens at
#theta = (3pi)/4#
and
#theta = (7pi)/4#

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Answer 2

To evaluate arctan(-1) using a unit circle, we consider the definition of arctan as the angle whose tangent is equal to the given value. Since tangent is negative in the second and fourth quadrants of the unit circle, we look for the angle in those quadrants where the tangent is -1.

In the second quadrant, the tangent is negative and equal to -1 at the angle of π - π/4, or 3π/4 radians.

In the fourth quadrant, the tangent is also negative and equal to -1 at the angle of -π/4, or 7π/4 radians.

Therefore, arctan(-1) can be evaluated as either 3π/4 or 7π/4 radians.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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