Inverse Tangent
tan^{1}
Tan^{1}
arctan
Arctan
The inverse function of tangent.
Basic idea: To find tan^{1} 1,
we ask "what angle has tangent equal to 1?" The answer
is 45°. As a result we say that tan^{1} 1
= 45°.
In radians this is tan^{1} 1
= π/4.
More: There are actually many angles that have
tangent equal to 1. We are really asking "what is the simplest,
most basic angle that has tangent equal to 1?" As before,
the answer is 45°.
Thus tan^{1} 1
= 45° or tan^{1} 1 = π/4.
Details: What is tan^{1} (–1)?
Do we choose 135°, –45°, 315° ,
or some other angle?
The answer is –45°.
With inverse tangent, we select the angle on the right half of the unit
circle having measure as close to zero as possible. Thus tan^{1} (–1)
= –45° or tan^{1} (–1) = –π/4.
In
other words, the range of tan^{1} is
restricted to (–90°, 90°) or .
Note: arctan
refers to "arc tangent", or the radian measure of the
arc on a circle corresponding to a given value of tangent.
Technical note: Since none of the six trig functions sine, cosine, tangent, cosecant, secant, and cotangent are onetoone,
their inverses are not functions. Each trig function can have its
domain restricted, however, in order to make its inverse a function.
Some mathematicians write these restricted trig functions and their
inverses with an initial capital letter (e.g. Tan or Tan^{1}).
However, most mathematicians do not follow this practice. This
website does not distinguish between capitalized and uncapitalized
trig functions.
See
also
Inverse
trigonometry, inverse
trig functions, interval notation
