**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:

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)

**Business: Reversion Value**

Input:

A = n: number of payments

B = I%: periodic interest rate (in decimal)

C = monthly rent

D = asking price

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

Asking Price, D = 250000

Result: 204337.26

Source:

Roger F. Farish and Elbert B. Greynolds, Jr., Ph.D., CPA

__Calculator Analysis for Business and Finance__Texas 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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