# “How will you evaluate the approximate value of tan 46 degrees?”

Came up with a bit of a strange method for an approximation of tan(46 degrees)

I would first note that 46 degrees is close to 45 degrees.

The tan(45 degrees) = 1 since it’s a 45 degree triangle and the opposite over the adjacent is going to give you 1.

46 degrees is slightly above 45 degrees, so the opposite side is going to be a bit bigger than the adjacent side.

Therefore it will be a bit over 1.

From there I’ll think about how much different it will be than tangent of 45 degrees.

Since you’re given that 1 degree 0.01745 radians,

I think about the opposite side getting a bit bigger and the adjacent side of the triangle (for the 46 degree right triangle) getting a bit smaller (equal amounts).

I call the change ∆. If we call the opposite y and the adjacent x, then the slightly changed sides would change by this amount ∆. And because tanx ~ x for small angles, I’m thinking about the change in the angle 46-45 = 1 degree as the angle, which is fairly small.

So for an approximation, something that I came up with that seems to work…..

tan(46 degrees) ~ (1+0.01745)/ (1-0.01745)

= 1.035519821

If I just type in tan(46 degrees), I get

=1.035530314

If I use more accuracy in the calculator for the radian equivalent of 1 degree, I get

tan(46)~ 1.035526642

Which is different by 3.67188 x 10^-6

This method seems to work even for larger changes in angle. For example

tan(50) ~  (1+0.01745*5)/ (1-0.01745*5)

= 1.191180498

On the calculator

tan(50) = 1.191753….

So a bit less accurate, but still accurate to the nearest hundredth.

Not quite sure why this works.