Question

Hi,

I have the following formula:

UEF=0.8165+0.2254tan^-1((Year-1967.5)/21.25)

Which is supposed to produce the following values of UEF:
UEF=1.06 for year = 2010
UEF=1.03 for year = 2000
UEF=1.00 for year = 1990
UEF=0.94 for year = 1980
UEF=0.85 for year = 1970
UEF=0.74 for year = 1960
(note that the above UEF values may not be quite right as I have taken them from a plot which was pretty low quality).

The problem is that when plugging various values for the variable "Year" into the formula, I am unable to replicate the above quoted UEF values that I took from the plotted graph of the relationship. I recieve the following answers which are obviously well out:

UEF=15.11 for year = 2010
UEF=13.62 for year = 2000
UEF=11.33 for year = 1990
UEF=7.68 for year = 1980
UEF=2.33 for year = 1970
UEF=-3.57 for year = 1960

I would be most gratful if anyone could offer an insight as to where I'm going wrong.

Best regards
Aaron

Answer 1

We have next formula:

$$UEF = 0.8165 + 0.2254 \tan^{-1} \left( \frac{Year - 1967.5}{21.25} \right)$$

Answer: We have an angel in radian. We mean that we use value of $\tan^{-1}(x)$ function in radians (not in degrees). Then we will have right answers.

Let's take year 2010:

$$\begin{array}{l} UEF = 0.8165 + 0.2254 \tan^{-1} \left( \frac{2010 - 1967.5}{21.25} \right) = \\ = 0.8165 + 0.2254 \tan^{-1} (2) = \\ = 0.8165 + 0.2254 \cdot 1.07 = 1.06. \end{array}$$

In the same way we will have:

For year = 2000: $UEF = 0.8165 + 0.2254 \tan^{-1} \left( \frac{2000 - 1967.5}{21.25} \right) = 1.03$;

For year = 1990: $UEF = 0.8165 + 0.2254 \tan^{-1} \left( \frac{1990 - 1967.5}{21.25} \right) = 1.00$;

For year = 1980: $UEF = 0.8165 + 0.2254 \tan^{-1} \left( \frac{1980 - 1967.5}{21.25} \right) = 0.94$;

For year = 1970: $UEF = 0.8165 + 0.2254 \tan^{-1} \left( \frac{1970 - 1967.5}{21.25} \right) = 0.84$;

For year = 1960: $UEF = 0.8165 + 0.2254 \tan^{-1} \left( \frac{1960 - 1967.5}{21.25} \right) = 0.74$.