What is the value of tan inverse of 1?

The inverse of a tangent function is arctan or inverse tan or atan. i.e., tan-1 = arctan. The value of tangent function varies from -∞ to +∞. Enter the value into the calculator, the atan of the entered value will be displayed in either degrees or radians.

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Formula

Formula:

t = tan-1(x) Where, t = Inverse Tangent x = Value

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The inverse tan of 1, ie tan-1 (1) is a very special value for the inverse tangent function. Remember that tan-1(x) will give you the angle whose tan is x . Therefore, tan-1 (1) = the angle whose tangent is 1. It's also helpful to think of tangent

The Value of the Inverse Tan of 1

As you can see below, the inverse tan-1 (1) is 45° or, in radian measure, Π/4. It is helpful to think of tangent as the ratio of sine over cosine, ie:

. Therefore, tan(Θ) to equal 1, sin(Θ) and cos(Θ) must have the same value.

So-When do sine and cosine have the exact same value?

The answer is at

and, of course, sine these trig functions are cyclical you can generalize

My answer's a bit more jaded than the other answer.

Trig students are only expected to know "exactly" the trig functions of two triangles, 30/60/90 and 45/45/90. It seems insane to have a whole field about just two triangles, but once you accept it trig becomes easier.

So you only need to know two triangles, but you need to know them in each quadrant, or at least be able to figure them out.

I really don't like the notation #tan^{-1}(x)# for #arctan(x)#. I prefer the small letter #arctan(x)# to be multivalued, reserving #text{Arc}text{tan}(x)# for the principal value.

#arctan(x) = text{Arc}text{tan}(x) + 180^circ k quad # integer #k#, or,

#arctan(x) = text{Arc}text{tan}(x) + kpi quad # in radians.

We'll "solve" both #text{Arc}text{tan}(-1) and arctan(-1).#

There's not a lot of solving involved. The expression #arctan(1)# means all the angles whose tangents are #1#. Tangents are slopes so that's all angles whose rays have a slope of #1#. That's one of our two triangles, #45^circ # and #180^circ+45^circ=225^circ# plus their coterminal brethren.

We have #arctan(-1).# The negative slope means we're after the analogous triangles in the second and fourth quadrants. That's #-45^circ# in the fourth quadrant and #135^circ# in the second.

#arctan(-1) = -45^circ + 180^circ k quad # integer #k#

The principal value for all these inverse functions are the continuous part which includes the first quadrant. Tangent blows up at #90^circ# so that's #-90^circ# to #90^circ.#

#text{Arc}text{tan}(-1) = -45^circ #

In radians

#arctan(-1) = -pi/4 + k pi quad # integer #k#

#text{Arc}text{tan}(-1) = -pi/4 #