The inverse trigonometric functions.

This week, we have been learning how to find all the trig functions of an angle if you only know sine, cosine or tangent.  Besides that, given any of these values we can find the angle itself by using the inverse trig function buttons.  On the TI-30X II s, the 2nd button is the blue button at the upper left hand corner.  We have already used it for square roots because the square root function is [2nd][x²].  The inverse trig functions of [sin], [cos], and [tan] are [2nd][sin], [2nd][cos] and [2nd][tan], respectively.

Almost none of the "nice" values of sine and cosine, the ones that are rational, give us "nice" angles.  The exception is ½. sin60° = cos30° = ½.

Here are some of the easy fractions for sine and the angles they represent, rounded to as close as the calculator will give them.

If sin alpha = 1/3, alpha  = 19.47122063°
If sin alpha = 2/3, alpha = 41.8103149°
If sin alpha =1/4, alpha = 14.47751219°
If sin alpha =3/4, alpha = 48.59037789°
If sin alpha =1/5, alpha = 11.53695903°
If sin alpha =2/5, alpha = 23.57817848°
If sin alpha =3/5, alpha = 36.86989765°

If sin alpha =4/5, alpha = 53.13010235°

Again, these are not exact.  The actual numbers are irrational.

One useful property of the inverse tangent law is that if we know tan alpha = x, then tan (90° - alpha) = 1/x.  This is to say if we have a line with slope 1/9 and we find the angle measure is beta, then the angle with slope 9 is (90° - beta).  In this particular case the approximate numbers are
tan 6.340191746° = 1/9 and
tan 83.659808254° = 9

Find the following angles for the given values of tangent, rounded to the nearest thousandth of a degree.  Answers in the comments.

a) If tan alpha = 2, alpha  =
b) If tan alpha = 3, alpha  =
c) If tan alpha = 4, alpha  =
d) If tan alpha = 5, alpha  =
e) If tan alpha = 6, alpha  =
f) If tan alpha = 7, alpha  =