How do you evaluate the inverse function by sketching a unit circle, locating the correct angle, and evaluating the ordered pair on the circle for: tan^-1 (0)tan1(0) and csc^-1 (2)csc1(2)?

1 Answer
Aug 4, 2014

The trigonometric functions ("sin"sin, "cos"cos, "tan"tan) all take angles as their arguments, and produce ratios. (remember SOHCAHTOA)

The inverse trigonometric functions ("arcsin"arcsin, "arccos"arccos) take ratios as their arguments, and produce the corresponding angles.

Let us take a look at a unit circle diagram:
Unit circle diagram with radiusUnit circle diagram with radius

rr is the radius of the circle, and it is also the hypotenuse of the right triangle.

We will start with arctan 0arctan0. First, we know that the tangent of an angle equals the ratio between the opposite side and the adjacent side. And, we know that the arc tangent function takes a ratio of this form, and produces an angle. Since 00 is our arc tangent's argument, then it must be equal to the ratio:

y/x = 0yx=0.

Clearly, this statement can only be true if y = 0y=0. And if y= 0y=0, then thetaθ must also be 00.

So,

arctan 0 = 0arctan0=0.

Let us move on to "arccsc"(2)arccsc(2).

Well, the cosecant of an angle is the inverse of its sine. In other words,

csc theta = 1/sin thetacscθ=1sinθ.

We know that sine gives a ratio between the opposite side and the hypotenuse. So, the cosecant function therefore gives a ratio between the hypotenuse and the opposite side. And, if the arc-cosecant takes this ratio as an argument, and gives the angle, then we know that 22 must be the ratio between the hypotenuse and the opposite side.

2 = r/y2=ry

This is more conveniently written as:

2y = r2y=r

Or, alternatively as:

y = 1/2 ry=12r

What this tells us is that for our angle thetaθ to equal the "arccsc"arccsc of 22, we need a right triangle whose hypotenuse is twice the length of its opposite leg.

And, elementary geometry tells us that this is precisely what occurs in a 30-60-90 triangle.

If r = 2yr=2y, then x = ysqrt(3)x=y3. Therefore, thetaθ is equal to 3030 degrees, or pi/6π6.