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# Casio fx-3650P: Chebyshev Polynomials, Roots of a Complex Number, Reversion Value, Mach Speed, Trapezoid Perimeter

Casio fx-3650P: Chebyshev Polynomials, Roots of a Complex Number, Reversion Value, Mach Speed, Trapezoid Perimeter
Chebyshev Polynomials of the First Kind
Values of the Chebyshev polynomials can calculated by the following:
T_n (x):
cos (n acos x) if |x| ≤ 1
cosh (n acosh x) if x ≥ 1
(-1)^n cosh (n acosh(-x)) if x ≤ 1
Inputs: x, y (n) order. The program assumes y is an integer.
Program (74 bytes):
? → X : ? → Y : X > 1 ⇒ Goto 1 :
-1 > X ⇒ Goto 2 : cos ( Y cos¹ X ) → A : Goto 3 :
Lbl 1 : cosh ( Y cosh¹ X ) → A : Goto 3
Lbl 2 : (-1)^Y * cosh ( Y cosh¹ X ) → A :
Lbl 3: A
Examples
T_Y(X)
X: -3, Y: 5, Result: -3363
X: 0.3, Y: 5, Result: 0.99888
X: 3, Y: 5, Result: 3363
Roots of a Complex Number
This program calculates the roots of a complex number (x + yi)^a, which is calculated as:
r^(1/n) * ( cos((θ + 2 π k)/n) + i*sin((θ + 2 π k)/n) )
where k = 0 to n-1
r =√(x^2 + y^2)
θ = angle(x + yi) = atan(y/x)
Program (66 bytes):
? → X : ? → Y : ? → A : 0 → M :
Rad : Pol(X,Y) : A x√ X → B :
Y → D : Lbl 1 : M ⊿ Rec(B, (D + 2 π M )/A) ⊿
Y ⊿ 1 M+ : M ≠ A ⇒ Goto 1 : M
The program stops when M = A.
Example:
(-2 + 4i)^(1/4)
X: -2, Y: 4, A: 4
(Fix 4 setting)
M: 0, X: 1.2701, Y: 0.7082
M: 1, X: -0.7082, Y: 1.2701
M: 2, X: -1.2701, Y: -0.7082
M: 3, X: 0.7082, Y: -1.2701
M: 4 (end)
Input:
A = n: number of payments
B = I%: periodic interest rate (in decimal)
C = monthly rent
Note: For monthly payments, divide the annual interest rate by 1200.
Program (55 bytes):
Fix 2 : ? → A : ? → B : ? → C : ? → D :
-C (1 + B) ( (1 + B)^A -1 ) ÷ B → X :
X + D (1 + B)^A → X : X
Example:
5 years of monthly payments, A = 60
Interest rate, B = 6/1200
Monthly Rate, C = 1895

Result: 204337.26
Source:
Roger F. Farish and Elbert B. Greynolds, Jr., Ph.D., CPA Calculator Analysis for Business and FinanceTexas Instruments Incorporated 1978 ISBN 0-07-063757-1
Mach Speed
This program calculates the mach speed, a ratio of the speed of sound given:
A = the altitude of the object, such as a vehicle (in feet)
B = temperature (°F)
C = velocity of the vehicle (mph)
Equations used:
Mach Number = velocity of the vehicle / speed of sound
where Speed of Sound = 44100 / 1519 * √( 459.7 + B – 3.57 * A / 1000)
Program (52 bytes):
? → A : ? → B : ? → C :
44100 ÷ 1519 * √ ( 459.7 + B – 3.57 A ÷ 1000 ) → D :
C ÷ D → M
A lapse rate of -3.57° per 1000 ft altitude is assumed.
Example:
Input:
A: altitude = 2500 ft
B: temperature = 63 °F
C: velocity = 795 mph
Result:
mach speed: 1.208091501
Source:
Mach Number and Airspeed vs Altitude” Granite Island Group (Technical Surveillance Counter Measures) http://www.tscm.com/mach-as.pdf Retrieved January 26, 2019
Perimeter of a Trapezoid
P + A + B + C * (1/sin X + 1/sin Y)
Program (38 bytes):
Deg : ? → A : ? → B : ? → C : ? → X : ? → Y :
A + B + C ( 1 ÷ sin X + 1 ÷ sin Y )
Example:
A = 227.6
B = 305.4
C = 10.5
X = 86°
Y = 5°
Result: 663.999629
Eddie
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