Casio fx-CG 50 CORDIC Simulation: Approximating Sine and Cosine of Angles

Casio fx-CG 50 CORDIC Simulation:  Approximating Sine and Cosine of Angles

Introduction - How Computers Calculate Mathematical Functions

First developed by Jack E. Volder, the Coordinate Rotation Digital Computer, better known as CORDIC, is an algorithm used to calculate many mathematical functions, including trigonometric functions, logarithms, exponentials, and hyperbolic functions.     CORDIC is a fundamental algorithm, which variants of CORDIC are used in computers and calculators.

Today's focus will be a calculating sines and cosines of angles.  The steps will be detailed in the next sections.  

CORDIC1:  Series of Arctangents

Let Θ be the angle.   The first step is to build Θ as additions and subtractions of terms of arctan(1 ÷ (2^n)), starting at n = 0 and stopping at the required accuracy.  Let A be the approximation.  

Examples: 

Θ = 45° requires only one term:

A = 45° = arctan 1

Θ ≈ 71.56505118°

 A = 71.56505118° = arctan 1 + arctan 1/2

Θ ≈ 32.47119229°

A = 32.47119229° = arctan 1 - arctan 1/2 + arctan 1/4

A series to recreate Θ = 30° takes 32 terms of ±arctan (1 ÷ 2^n) from n = 0 to n = 31.   (8 decimal places)  This is known as coordinate rotation.

The program CORDIC1 determines the number of terms need to created to obtain Θ.   Accuracy in this code is set to 5 decimal places, but can be adjusted (see the line in blue).  

Casio fx-CG 50 Program Code:  CORDIC1

Deg

"ANGLE"?->Θ

0->I

0->A

Lbl 0

tan^-1 (2^(-I))->T

If A<Θ

Then 

A+T->A

Else 

A-T->A

IfEnd

I+1->I

Abs (A-Θ)>1×10^-5=>Goto 0

ClrText

Blue Locate 1,3,"Θ: "

Blue Locate 5,3,Θ

Red Locate 1,4,"A: "

Red Locate 5,4,A

Green Locate 1,7,I

Notes:

-> is the store arrow →

=> is the jump command ⇒

For reference:

arctan 1 = 45°

arctan 1/2 ≈ 26.56505118°

arctan 1/4 ≈ 14.03624347°

arctan 1/8 ≈ 7.125016349°

arctan 1/16 ≈ 3.576334375°

arctan 1/32 ≈ 1.789910608°

CORDIC2:  Calculating Sine and Cosine

Imagine the coordinate (cos Θ, sin Θ) on a unit circle.  A unit circle is a circle with radius of length 1 and center located at the origin (0,0).   

Set the initial angle at 0°.  Then x = cos 0° = 1 and y = sin 0° = 0.   The initial vector is set to be [ [ cos 0° ] [ sin 0° ] ] = [ [ 1 ] [ 0 ] ].

Approximate Θ in terms of arctangent (1 ÷ (2^n)), starting at n = 0.  

Direction of Rotation

Set σ as the direction of rotation.    

A rotation is positive if we add arctan(1 ÷ (2^n)) to A.    For a positive rotation, set σ_i = +1.   

A rotation is negative if we subtract arctan(1 ÷ (2^n)) to A.    For a negative rotation, set σ_i = -1.   

For example:

 Θ ≈ 32.47119229°

A = 32.47119229° = arctan 1 - arctan 1/2 + arctan 1/4

Then:  σ_0 = 1 (positive rotation), σ_1 = -1 (negative rotation), σ_2 =  1 (positive rotation).  

Multiplying Factor

The number of iterations is also used to determine the required multiplication factor:

n = Π( 1 ÷ √(1 + 2^(-2 × I)), I = 0 to I = terms needed - 1)

For the example:

 Θ ≈ 32.47119229°

Needed 3 iterations, i_0 to i_2.  

Then:

n = 1 ÷ √(1 + 2^(-2 × 0)) × 1 ÷ √(1 + 2^(-2 × 1)) × 1 ÷ √(1 + 2^(-2 × 2))

= 4 × √170 ÷ 85

≈ 0.6135719911

Calculating the Next Iteration

The next iteration for x and y are:

x_i+1 = x_i - 2^(-i) × σ_i × y_i

y_i+1 = 2^(-i) × σ_i × x_i + y_i

When A is sufficiently near or equal to Θ, the cosine and sine are approximated as:

cos(A) ≈ n × x_final

sin(A) ≈ n × y_final

For the example:

 Θ ≈ 32.47119229°

The approximated angle, in this case, happens to be exact angle, A = Θ.  And,

cos(A) ≈ 0.8436614877 

sin(A) ≈ 0.5368754922

For more details, please check out resources in the Sources section.   I particularly like Oxford's A Very Short Introduction series.  

Casio fx-CG 50 Program Code:  CORDIC2

This algorithm is adopted from the Python example (see Wikipedia article).  

Deg

"ANGLE"?->Θ

0->I

0->A

1->X

0->Y

1->N

Lbl 0

tan^-1 (2^(-I))->T

If A<Θ

Then 

A+T->A

1->S

Else 

A-T->A

-1->S

IfEnd

N÷√(1+2^(-2*I))->N

X-S*2^(-I)*Y->P

Y+X*S*2^(-I)->Q

P->X

Q->Y

I+1->I

Abs (A-Θ)>1×10^-5=>Goto 0

X*N->X

Y*N->Y

ClrText

Blue Locate 1,3,"Θ: "

Blue Locate 5,3,Θ

Red Locate 1,4,"A: "

Red Locate 5,4,A

Black Locate 1,5,"cos :"

Black Locate 7,5,X

Black Locate 1,6,"sin :"

Black Locate 7,6,Y

Green Locate 1,7,I

CORDIC3:  Doing it without a Arctangent function

In reality, most of the time the arctangent function also has to be approximated.   There are many ways to approximate, which varying accuracy.   I used the fx-CG50's statistics mode to come up with a regression equation with the following lists:

x_list = sequence of 1÷(2^i) from i = 0 to i = 39 

y_list = sequence of arctan(1÷(2^i)) from i = 0 to i = 39

Of the regression models the fx-CG50 offers, the best regression model is quartic regression (4th-order polynomial):

y ≈ 9.43597784 × x^4 - 22.116232 × x^3 + 0.40116676 × x^2 + 57.2790727 × x + 1.4558 × 10^-5

Remember that I am working with degree angle measurement.  

Casio fx-CG 50 Program Code:  CORDIC3

Deg

"ANGLE"?->Θ

0->I

0->A

1->X

0->Y

1->N

Lbl 0

2^(-I)->K

9.43597784917955K^(4)-22.1162323028173K^(3)+

0.401166766193516K^2+57.2790727477367K+

1.45584715586876×10^-5->T

If A<Θ

Then 

A+T->A

1->S

Else 

A-T->A

-1->S

IfEnd

N÷√(1+2^(-2*I))->N

X-S*2^(-I)*Y->P

Y+X*S*2^(-I)->Q

P->X

Q->Y

I+1->I

Abs (A-Θ)>1×10^-5=>Goto 0

X*N->X

Y*N->Y

ClrText

Blue Locate 1,3,"Θ: "

Blue Locate 5,3,Θ

Red Locate 1,4,"A: "

Red Locate 5,4,A

Black Locate 1,5,"cos :"

Black Locate 7,5,X

Black Locate 1,6,"sin :"

Black Locate 7,6,Y

Green Locate 1,7,I

For the example:

 Θ ≈ 32.47119229°

Approximating Θ within 8 decimal places (10^-5) yields these results:

A:  32.47118598

cos A:  0.8436683456

sin A:  0.5368647154

24 terms used

Sources

"CORDIC"  Wikipedia.   Last Edited January 11, 2024.   Accessed January 12, 2024.  https://en.wikipedia.org/wiki/CORDIC

Brummelen, Glen Van.  Trigonometry:  A Very Short Introduction  Oxford University Press: Oxford, United Kingdom.  2020.  ISBN 978-0-19-881431-3

Eddie

All original content copyright, © 2011-2024.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

March Calculus Madness Sweet Sixteen - Day 13: Some Double Integrals

 ------------

Welcome to March Calculus Madness!

------------

Double Integration Time

(I) 

∫ ∫ sin(a*x + b*y) dx dy

= ∫ -1/a * cos(a*x + b*y) + C1 dy

= -1/(a*b) * sin(a*x + b*y) + C1*y + C2

(II) 

∫ ∫ sin(a*x + b*y) dy dx

= ∫ -1/b * cos(a*x + b*y)+ C1 dx

= -1/(a*b) * sin(a*x + b*y) + C1 * x + C2

For (I) and (II) to equal,  x = y

(III)

∫ ∫ e^(a*x + b*y) dx dy

= ∫ 1/a * e^(a*x + b*y)+ C1 dy

= 1/(a*b) * e^(a*x + b*y) + C1 * y + C2

(IV)

∫ ∫ e^(a*x + b*y) dy dx

= ∫ 1/b * e^(a*x + b*y) + C1  dx

= 1/(a*b) * e^(a*x + b*y) + C1 * x + C2

For (III) and (IV) to be equal, x = y

Eddie

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

March Calculus Madness Sweet Sixteen - Day 7: ∫ 1/(1 + cos x) dx and ∫ 1/(1 + sin x) dx

 ------------

Welcome to March Calculus Madness!

------------

∫ 1/(1 + cos x) dx and ∫ 1/(1 + sin x) dx

To tackle these integrals, we will make use of the following derivatives and trig identities:

d/dx tan x = sec^2 x dx

d/dx cot x = -csc^2 x dx

d/dx csc x = -cot x * csc x

d/dx sec x = tan x * sec x

sin^2 x + cos^2 x = 1

cot x = 1/tan x,  csc x = 1/sin x,  sec x = 1/cos x

∫ 1/(1 + cos x) dx

= ∫ 1/(1 + cos x) * (1 - cos x)/(1 - cos x) dx

= ∫ (1 - cos x)/(1 - cos^2 x) dx

= ∫ (1 - cos x)/sin^2 x dx

= ∫ csc^2 x - cot x * csc x dx

= -cot x + csc x + C

∫ 1/(1 + sin x) dx

= ∫ 1/(1 + sin x) * (1 - sin x)/(1 - sin^2 x) dx

= ∫ (1 - sin x)/(1 - sin^2 x) dx

= ∫ (1 - sin x)/(cos^2 x) dx

= ∫ sec^2 x - tan x * sec x dx

= tan x - sec x + C

Note:  CAS systems in graphing calculators will default to using sin x, cos x, and tan x.  

Eddie

All original content copyright, © 2011-2022.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

An Alternative Way of Finding the Angle in a Rectangular to Polar Conversion

An Alternative Way of Finding the Angle in a Rectangular to Polar Conversion

Welcome to a special Monday edition of Eddie’s Math and Calculator Blog. 

The Traditional Method

Often we are required to find polar coordinates of a given point (x,y).   Finding the radius, r, is fairly simple:

r = √(x^2 + y^2) 

When talking about complex numbers, r represents the absolute value of x + yi where i = √-1.

Finding the angle, θ, often uses the formula:

θ = atan(y/x)

In complex numbers, θ represents the argument (arg) function.

On a scientific calculator the range of the arctangent function is ( -90°, 90° ).  (open interval).   In finding the true angle, adjustments will be required:

Let a = atan(y/x). Then: 

Quadrant I (x and y are both positive):  θ = a

Quadrant II (x is negative, y is positive): θ = a + 180°

Quadrant III (x and y are both negative):  θ = a - 180°

Quadrant IV (x is positive, y is negative):  θ = a 

If you are working with radian angle measures, know that 90° = π/2, and 180° = π.

This does not take into consideration situations where either x or y is 0:

If x > 0 and y = 0:  θ = 0°

If x = 0 and y > 0:  θ = 90°

If x < 0 and y = 0:  θ = 180°

If x = 0 and y < 0:  θ = -90°

Is there a shorter way to calculate θ?  

The Vector Method

Consider the point (x, y) as a vector [x, y].   Now draw another vector [x, 0].  In a regular Cartesian coordinate system, angles are measured from the x-axis counter clockwise.  

Let a and b represent two vectors.  Then the angle between two vectors are:

cos θ = (a ● b) / ( ||a|| ||b|| ) = dot(a,b) / ( norm(a) * norm(b) )

with:

dot(a,b) = a1 * b1 + a2 * b2

norm(a) = √(a1^2 + a2^2)

norm(b) = √(b1^2 + b2^2)

Let a = [x, y] and b = [x, 0].  Then:

cos θ = (x^2) / (√(x^2 + y^2) * √(x^2))

cos θ = (x^2) / (√(x^2 + y^2) * x)

cos θ = x / √(x^2 + y^2)

θ = acos( x / √(x^2 + y^2) )

The range of the arccosine function of a calculator is [ 0°, 180° ].  

If y < 0, the angle would be measured clockwise, and therefore I would make the adjustment:

θ = -acos( x / √(x^2 + y^2) )

In summary:

If y ≥ 0, θ = acos( x / √(x^2 + y^2) )

If y < 0, then θ = -acos( x / √(x^2 + y^2) )

Examples:

Find the angle, in degrees, in a rectangular to polar conversions:

Quadrant I  (2, 4):  y ≥ 0:   θ = acos( 2 / √(2^2 + 4^2) ) ≈ 63.43494882°

Quadrant II (-2, 4):  y ≥ 0:    θ = acos( -2 / √((-2)^2 + 4^2) ) ≈ 116.5650512°

Quadrant II (-2, -4):  y < 0:  θ = -acos( (-2) / √((-2)^2 + (-4)^2) ) ≈ -116.5650512°

Quadrant IV (2, -4):  y < 0:  θ = -acos( 2) / √(2^2 + (-4)^2) ) ≈ -63.43494882°

Vector Method for Navigation

In navigation, angles start from true North (up) and rotate clockwise towards East (right).  Angles are measured from 0° to 360°.

Use the vectors [0, N] and [E, N], then the angle between these vectors are:

If E ≥ 0,  θ = acos( N / √(E^2 + N^2))

If E < 0, θ = 360° - acos( N / √(E^2 + N^2))

Note:  This is the first blog entry that I have typed on Google Docs.  I have been using Windows app Wordpad for the last three years.   I am testing Google apps as I am considering buying a Chromebook.

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

HP 71B: Sign "Graph" of Trig Functions

 HP 71B: Sign "Graph" of Trig Functions

Introduction

The program SGNTRIG builds a 22 character binary string that depends on the function:

g(x) = 

1  if  sgn(t(x/c)) > 1

0  if  sgn(t(x/c)) ≤ 0

where:

c is a scaling factor, 

sgn(x) is the sign function,  and

t(x) represents one of three trigonometric functions, sin(x), cos(x), or tan(x).   Angles are assumed to be radians.

If we set to scale to c = 1, then g(x) takes the integer values from 1 to 22.  

The HP 71B builds the result to the string S$.   This is an aid to visualize at least some of the graph.   When the bit is 1, the graph is above the x-axis, otherwise, the graph is either on or below the x-axis.  The program presented can be expanded or modified to explore other functions.

The resulting string is a "psuedo-graph".  

Below is a graphical representation of g(x) (an HP Prime is used to graph g(x)).

HP 71B Program SGNTRIG

Size:  270 bytes

100 DESTROY S,I,N,E,C

105 S$=""

110 RADIANS

115 INPUT 'SCALE? ';C

120 DISP "S:SIN C:COS T:TAN"

125 E$=KEY$

130 IF E$="S" OR E$="C" OR E$="T" THEN 200 ELSE 120

200 FOR I=1 TO 22

205 IF E$="S" THEN N=SGN(SIN(I/C))

210 IF E$="C" THEN N=SGN(COS(I/C))

215 IF E$="T" THEN N=SGN(TAN(I/C))

220 IF N=1 THEN N=1 ELSE N=0

225 S$=S$&STR$(N)

230 DISP S$ @ BEEP 589,0.1 @ WAIT 0.2

235 NEXT I

240 DISP S$ @ BEEP 661,2

Notes:

Line 105:  creates a blank string, which is allowed on the HP 71B

Line 125:  The KEY$ function calls for a key input. S for sine, C for cosine, and T for tangent.

Lines 205 to 215: determine which trigonometric function to use

Line 220:  Sets the character to 0 if the sign function returns 0 or -1

Line 225:  &, the ampersand symbol concatenates two strings

Lines 230 and 240, respectively:  589 Hz is middle D, 661 Hz is middle E

Example

Below are results for each of the trigonometric functions when scale is set to 1.  

YouTube video: https://www.youtube.com/watch?v=GL6urOGvRGY&feature=youtu.be

Eddie

All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Sines and Cosines: Adding and Subtracting Angles

 Sines and Cosines:   Adding and Subtracting Angles

Note:   

π/2 radians = 90°,   π radians = 180°

Sine

sin(x + π/2) = sin(x) cos(π/2) + cos(x) sin(π/2) = cos(x)

sin(x - π/2) = sin(x) cos(π/2) - cos(x) sin(π/2) = -cos(x)

sin(π/2 - x)  = sin(π/2) cos(x) - cos(π/2) sin(x) = cos(x)

sin(x + π) = sin(x) cos(π) + cos(x) sin(π) = -sin(x)

sin(x - π) = sin(x) cos(π) - cos(x) sin(π) = -sin(x)

sin(π - x)  = sin(π) cos(x) - cos(π) sin(x) = sin(x)

Cosine

cos(x + π/2) = cos(x) cos(π/2) - sin(x) sin(π/2) = -sin(x)

cos(x - π/2) = cos(x) cos(π/2) + sin(x) sin(π/2) = sin(x)

cos(π/2 - x) = cos(π/2) cos(x) + sin(π/2) sin(x) = sin(x)

cos(x + π) = cos(x) cos(π) - sin(x) sin(π) = -cos(x)

cos(x - π) = cos(x) cos(π) + sin(x) sin(π) = -cos(x)

cos(π - x) = cos(π) cos(x) + sin(π) sin(x) = -cos(x)

Eddie

All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

TI-84+ and Casio (fx-CG 50) Micropython: Simplifying A sin x + B cos x To r sin (x + θ)

TI-84+ and Casio (fx-CG 50) Micropython:  Simplifying  A sin x + B cos x To r sin (x + θ)

Introduction

Simplify the expression:

(I)   a * sin x + b * cos x

to an expression that involves only one trigonometric function. 

The first task is to multiply the expression by √(a^2 + b^2)/√(a^2 + b^2). (refer to Source)  This results as:

(II) a* √(a^2 + b^2)/√(a^2 + b^2) * sin x + b * √(a^2 + b^2)/√(a^2 + b^2) * cos x

Refer to the triangle diagram below and it will should become clear why it is desirable to multiply by √(a^2 + b^2)/√(a^2 + b^2). 

With:

(III)
 √(a^2 + b^2) * a/√(a^2 + b^2) * sin x +  √(a^2 + b^2) * b/√(a^2 + b^2) * cos x
=  √(a^2 + b^2) * cos θ * sin x +  √(a^2 + b^2) * sin θ * cos x
=  √(a^2 + b^2) * (cos θ * sin x +  sin θ * cos x)

With the use of the trigonometric identity:
sin(x + y) = sin x * cos y + cos x * sin y

This becomes:
(IV)
 =  √(a^2 + b^2) * sin(x + θ)

Let's make some observations.

1.  The angle θ dependents on what quadrant the point (a,b) is. 

2.  Taking into consideration of where the point (a,b) is and the expression √(a^2 + b^2), the task of finding the coefficients of reduced formula for a * sin x + b* cos x can be achieved by calculating a rectangular to polar conversion of the point (a,b). 

Hence: (IV) becomes:

(V)
 a * sin x + b * cos x =  r * sin(x + θ)

where
r = √(a^2 + b^2)  = abs(a + bi)
θ = atan(b/a) = atan2(b,a) = arg(a + bi)


TI 84 Plus Program:  SCTOSIN

"EWS 2018-11-12"
a+bi
Disp "A*sin(X)+B*cos(X)", "=R*sin(X+θ)"
Prompt A,B
abs(A+B*i)→R
angle(A+B*i)→θ
Disp R,"sin(X+", θ, ")"

Casio Micropython (fx-CG 50) Script scotosin.py

import math
print("a*sin(x)+b*cos(x)")
print("=r*sin(x+t)")
a=float(input("a:"))
b=float(input("b:"))
r=math.sqrt(a**2+b**2)
t=math.atan2(b,a)
print(" ")
print(r)
print("*sin(x+")
print(t,")")

Examples
(Radians mode assumed)

Example 1:
4 * sin x + 2 * cos x = 4.472135955 * sin(x + 0.463647609)

Input:
a = 4
b = 2

Output:
r = 4.472135955
θ = 0.463647609



Example 2:
3 * sin x - 5 * cos x = 5.803951895 * sin(x - 1.030376827)

Input:
a = 3
b = -5

Output:
r = 5.803951895
θ = -1.030376827


Source:
Dugopolski, Mark  Trigonometry Addison Wesley: Boston 2003 pp 211-212  ISBN 0-201-70338-6

Eddie

All original content copyright, © 2011-2018.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.  Please contact the author if you have questions.

HP 12C Platinum: Cosine of an Angle

HP 12C Platinum:  Cosine of an Angle

Introduction

This is a fairly short program of an angle.  The angle is in radians.  This is adapted from the TI-84 Plus COSRAD program from the book Inside Your Calculator (see Source section below).

HP 12C Platinum Program:  Cosine of an Angle

The angle is entered in radians.  This program can be adopted for the regular HP 12C.

Step	Key Code	Key	Comment
001	44, 0	STO 0	Begin program
002	2	2
003	21	y^x	You could use x^2 on the Platinum
004	4	4
005	2	2
006	9	9
007	4	4
008	9	9
009	6	6
010	7	7
011	2	2
012	9	9
013	6	6
014	10	÷
015	44, 1	STO 1
016	1	1	Set counter
017	6	6
018	44, 2	STO 2
019	4	4	Loop begins
020	45, 1	RCL 1
021	30	-
022	45, 1	RCL 1
023	20	*
024	44, 1	STO 1
025	1	1
026	44, 30, 2	STO- 2	Decrease counter
027	45, 2	RCL 2
028	43, 35	x=0
029	43, 33, 031	GTO 031
030	43, 33, 019	GTO 019
031	1	1
032	45, 1	RCL 1
033	2	2
034	10	÷
035	30	-
036	43, 33, 000	GTO 000

Examples (FIX 6 setting):

cos 0.445 ≈ 0.902611

cos 2 ≈ -0.416147

cos -4.180 ≈ -0.507593

cos π/6 ≈ cos 0.523599 ≈ 0.866025

Source:

Gerald R. Rising.  Inside Your Calculator:  From Simple Programs to Significant Insights  Wiley-Interscience, John Wiley & Sons, Inc: Hoboken, New Jersey.  2007. ISBN 978-0-470-11401-8

Eddie

All original content copyright, © 2011-2018.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.  Please contact the author if you have questions.