Easy Trig Approximations

y = tan(x) and y = atan(x)  y = arctan(x) for any angle from 0 to up to 90 degrees

Polynomial Approximations for tan(x) and arctan(x/y) arctan(opposite/hypoteneuse) with Ɵ<=90 degrees


atan(x/y) = (x/y) - (x/y)^3/3 + (x/y)^5/5 with atan(x/y) in radians

convert radians to degrees

Ɵ = arctan(x/y) =90(0.64(x/y) + (x/y)^2 + (x/y)^3)/(1 + (1.64)(x/y) + (1.64)*(x/y)^2 + (x/y)^3)

y = arctan(x) = 90(0.64(x + x^2 + x^3)/(1 + (1.64)x + (1.64)x^2 + x^3)


In Microsoft Excel:

=DEGREES(ATAN(opposite/hypotenuse))

OR

=90*(0.64*(opposite/hypotenuse) + (opposite/hypotenuse)^2 + (opposite/hypotenuse)^3)/(1 + (1.64)*(opposite/hypotenuse) + (1.64)*(opposite/hypotenuse)^2 + (opposite/hypotenuse)^3)




For angles between 0 and 45 degrees

y=tax(x)= x + 1/3(x^3) + 2/15(x^5)  + 7/315 (x^7)

For angles between 45 and 90 degrees

y=tax(x)= -1/ ((90-x) + (90-x)^3/3 + 2/15(90-x)^5  + 7/315(90-x)^7) with x in radians

convert x to degrees

0<x<45 degrees:
y = tan(x) =(0.01745x)/(1- 0.0001x^2)

90>x>45 degrees:
y = tan(x) = 1/(0.01745(90-x) + (0.01382(90-x))^3+(0.0126(90-x))^5)

In Microsoft Excel:

TAN(RADIANS(angle with degrees) =((0.01745*(angle with degrees))/(1- 0.0001*(angle with degrees)^2)

TAN(RADIANS(angle with degrees) = 1/(0.01745*(90-(angle with degrees)) + (0.01382*(90-(angle with degrees)))^3+(0.0126*(90-(angle with degrees)))^5)