Trigonometry Reduction Formula and Solving Simple Arcsine and Arccosine Equations

Trigonometry Reduction Formula and Solving Simple Arcsine and Arccosine Equations

Some Background and Periodic Reduction Formulas

This blog will focus on angle measurement in degrees. For radians and grads, please use the appropriate measurement.

90° = π/2 rad = 100 grad

180° = π rad = 200 grad

sin 90° = 1, sin 180° = 0, sin 360° = 0, sin(-x) = -sin(x)

cos 90° = 0, cos 180° = -1, cos 360° = 1, cos(-x) = cos(x)

General Sums

sin(α + ß) = sin α * cos ß + sin ß * cos α

sin(α + 90°) = sin α * cos 90° + sin 90° * cos α = cos α

sin(α + 180°) = sin α * cos 180° + sin 180° * cos α = -sin α

sin(α + 360°) = sin α * cos 360° + sin 360° * cos α = sin α

sin(90° - α) = sin 90° * cos(-α) + cos 90° * sin(-α) = cos(-α) = cos α

sin(180° - α) = sin 180° * cos(-α) + cos 180° * sin(-α) = -1 * -sin α = sin α

cos(α + ß) = cos α * cos ß – sin α * sin ß

cos(α + 90°) = cos α * cos 90° – sin α * sin 90° = -sin α

cos(α + 180°) = cos α * cos 180° – sin α * sin 180° = -cos α

cos(α + 360°) = cos α * cos 360° – sin α * sin 360° = cos α

cos(90° - α) = cos 90° * cos(-α) – sin 90° * sin(-α) = -sin(-α) = sin α

cos(180° - α) = cos 180° * cos(-α) – sin 180° * sin(-α) = -1 * cos(-α) = -cos α

The Trigonometric Reduction Formula

a * sin(x) + b * cos(x) = √(a² + b²) * sin(x + Θ)

Let x = 90°:

a * sin(90°) + b * cos(90°) = √(a² + b²) * sin(90° + Θ)

a * 1 + b * 0 = √(a² + b²) * cos(Θ)

⇒ cos(Θ) = a ÷ √(a² + b²)

Let x = 180°:

a * sin(180°) + b * cos(180°) = √(a² + b²) * sin(180° + Θ)

a * 0 + b * -1 = √(a² + b²) * -sin(Θ)

-b = -sin(Θ) * √(a² + b²)

⇒ sin(Θ) = b ÷ √(a² + b²)

Then:

sin(Θ) = b ÷ √(a² + b²)

cos(Θ) = a ÷ √(a² + b²)

[sin(Θ) ÷ cos(Θ)] = [ b ÷ √(a² + b²) ] ÷ [ a ÷ √(a² + b²) ]

tan(Θ) = b ÷ a

Θ = arctan(b ÷ a)

Wikibooks uses the Algebraic Argument (see source):

a * sin(x) + b * cos(x)

= [ √(a² + b²) ÷ √(a² + b²) ] * [ a * sin(x) + b * cos(x) ]

= √(a² + b²) * ( a ÷ √(a² + b²) * sin(x) + b ÷ √(a² + b²) * cos(x) )

= √(a² + b²) * ( cos(Θ) * sin(x) + sin(Θ) * cos(x) )

= √(a² + b²) * sin(x + Θ)

Solving Simple Arcsine Equations

The calculator arcsine function gives: Domain: -1 ≤ x ≤ 1, Range: -90° ≤ Θ ≤ 90°

Note that for any angle x: sin(180° - x) = sin(x), sin(x) = sin(x ± 360°*z) (z is an integer)

Given n, solve for Θ:

n = sin(Θ) = sin(180° - Θ)

Base Solution 1:

n = sin(Θ)

⇒ Θ = arcsin(n)

Base Solution 2:

n = sin(180° - Θ)

arcsin(n) = 180° - Θ

⇒ Θ = 180° - arcsin(n)

Example:

0.67 = sin(Θ)

Base Solution 1: Θ = arcsin(0.67) ≈ 42.0670648025°

Base Solution 2: Θ = 180° - arcsin(0.67) ≈ 137.932935198°

Given n and α, solve for Θ:

n = sin(α + Θ)

Base Solution 1:

n = sin(α + Θ)

arcsin(n) = α + Θ

⇒ Θ = arcsin(n) – α

Base Solution 2:

n = sin(180° - (α + Θ))

n = sin(180° - α – Θ)

arcsin(n) = 180° - α – Θ

⇒ Θ = 180° - α – arcsin(n)

Example:

0.7757 = sin(Θ + 76°)

Base Solution 1: Θ = arcsin(0.7757) – 76° ≈ -25.1314549842°

Base Solution 2: Θ = 180° - 76° - arcsin(0.7757) = 104° - arcsin(0.7757) ≈ 53.131459842°

To get all the possible angles, add and subtract multiples of 360°.

Solving Simple Arccosine Equations

The calculator arccosine function gives: Domain: -1 ≤ x ≤ 1, Range: -90° ≤ Θ ≤ 90°

Note that for any angle x: cos(180° - x) = -cos(x), cos(x) = cos(x ± 360°*z) (z is an integer)

Given n, solve for Θ:

n = cos(Θ), n = cos(-Θ)

Base Solution 1:

n = cos(Θ)

⇒ Θ = arccos(n)

Base Solution 2:

n = cos(-Θ)

⇒ Θ = -arccos(n)

Example:

0.58 = cos(Θ)

Base Solution 1: Θ = arccos(0.58) ≈ 54.54945736°

Base Solution 2: Θ = -arccos(0.58) ≈ -54.54945736°

Given n and α, solve for Θ:

n = cos(α + Θ)

Base Solution 1:

n = cos(α + Θ)

arccos(n) = α + Θ

⇒ Θ = arccos(n) – α

Base Solution 2:

n = cos(-(α + Θ))

n = cos(-α – Θ)

arccos(n) = -α – Θ

-arccos(n) = α + Θ

⇒ Θ = -arccos(n) – α

Example:

0.6 = cos(35° + Θ)

Base Solution 1: Θ = arccos(0.6) – 35° ≈ 18.13012035°

Base Solution 2: Θ = -arccos(0.6) – 35° ≈ -88.13010235°

To get all the possible angles, add and subtract multiples of 360°.

Sources

Sterling, Mary Jane. Trigonometry for Dummies. 2^nd Edition. John Wiley & Sons, Inc. Hoboken, NJ. 2014. pp. 351-352. ISBN 978-1-118-82741-3

“Trigonometry/Simplifying a sin(x) + b cos(x)”. Wikibooks. January 2, 2024. Retrieved January 4, 2026. https://en.wikibooks.org/wiki/Trigonometry/Simplifying_a_sin(x)_%2B_b_cos(x)

Eddie

All original content copyright, © 2011-2026. 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.

Solving Simple Arcsine and Arccosine Equations

 Solving Simple Arcsine and Arccosine Equations

Angle Measure

This document will focus on angle measurement in degrees. For radians and grads, please use the appropriate measurement.

90° = π/2 rad = 100 grad

180° = π rad = 200 grad

Simple Arcsine Equations

The calculator arcsine function gives: Domain: -1 ≤ x ≤ 1, Range: -90° ≤ Θ ≤ 90°

Note that for any angle x: sin(180° - x) = sin(x), sin(x) = sin(x ± 360°*z) (z is an integer)

Given n, solve for Θ:

n = sin(Θ) = sin(180° - Θ)

Base Solution 1:

n = sin(Θ)

⇒ Θ = arcsin(n)

Base Solution 2:

n = sin(180° - Θ)

arcsin(n) = 180° - Θ

⇒ Θ = 180° - arcsin(n)

Example:

0.67 = sin(Θ)

Base Solution 1: Θ = arcsin(0.67) ≈ 42.0670648025°

Base Solution 2: Θ = 180° - arcsin(0.67) ≈ 137.932935198°

Given n and α, solve for Θ:

n = sin(α + Θ)

Base Solution 1:

n = sin(α + Θ)

arcsin(n) = α + Θ

⇒ Θ = arcsin(n) – α

Base Solution 2:

n = sin(180° - (α + Θ))

n = sin(180° - α – Θ)

arcsin(n) = 180° - α – Θ

⇒ Θ = 180° - α – arcsin(n)

Example:

0.7757 = sin(Θ + 76°)

Base Solution 1: Θ = arcsin(0.7757) – 76° ≈ -25.1314549842°

Base Solution 2: Θ = 180° - 76° - arcsin(0.7757) = 104° - arcsin(0.7757) ≈ 53.131459842°

To get all the possible angles, add and subtract multiples of 360°.

Solving Simple Arccosine Equations

The calculator arccosine function gives: Domain: -1 ≤ x ≤ 1, Range: -90° ≤ Θ ≤ 90°

Note that for any angle x: cos(180° - x) = -cos(x), cos(x) = cos(x ± 360°*z) (z is an integer)

Given n, solve for Θ:

n = cos(Θ), n = cos(-Θ)

Base Solution 1:

n = cos(Θ)

⇒ Θ = arccos(n)

Base Solution 2:

n = cos(-Θ)

⇒ Θ = -arccos(n)

Example:

0.58 = cos(Θ)

Base Solution 1: Θ = arccos(0.58) ≈ 54.54945736°

Base Solution 2: Θ = -arccos(0.58) ≈ -54.54945736°

Given n and α, solve for Θ:

n = cos(α + Θ)

Base Solution 1:

n = cos(α + Θ)

arccos(n) = α + Θ

⇒ Θ = arccos(n) – α

Base Solution 2:

n = cos(-(α + Θ))

n = cos(-α – Θ)

arccos(n) = -α – Θ

-arccos(n) = α + Θ

⇒ Θ = -arccos(n) – α

Example:

0.6 = cos(35° + Θ)

Base Solution 1: Θ = arccos(0.6) – 35° ≈ 18.13012035°

Base Solution 2: Θ = -arccos(0.6) – 35° ≈ -88.13010235°

To get all the possible angles, add and subtract multiples of 360°.

I hope you find this useful and helpful,

Eddie

All original content copyright, © 2011-2026. 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.

RPN: HP 11C: Transferring Between Bases (Common/Natural)

RPN: HP 11C: Transferring Between Bases (Common/Natural)

All algorithms were tested with the HP 11C.

Between the Exponential Function and Common-Antilog

e^α = 10^ß

Given α, what is ß?

ß = log(e^α)

RPN:

<input α>

e^x

LOG

Examples (Fix 6):

e^1.05 = 10^ß

ß ≈ 0.456009

e^(-2.2) = 10^ß

ß ≈ -0.955448

Given ß, what is α?

α = ln(10^ß)

RPN:

<input ß>

10^x

LN

Examples (Fix 6):

e^α = 10^5.4

α ≈ 12.433960

e^α = 10^0.366

α ≈ 0.842746

Between the Natural Logarithm and Common Logarithm

log α = ln ß

Given α, what is ß?

ẞ = exp(log α)

RPN:

<input α>

LOG

e^x

Examples (Fix 6):

log 17 = ln ß

ß ≈ 3.422766

log 317 = ln ß

ß ≈ 12.195405

Given ß, what is α?

α = 10^(ln ß)

RPN:

<input ß>

LN

10^x

Examples (Fix 6):

log α = ln 425

α ≈ 1,127,428.915

log α = ln 9.81

α ≈ 192.044677

Eddie

All original content copyright, © 2011-2026. 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 67 Programs… Almost 50 Years Later

 HP 67 Programs… Almost 50 Years Later

Both downloads are in PDF format. This is for use for the HP 67 and its emulators, or really almost any RPN scientific calculator. Enjoy!

Volume 1:

https://drive.google.com/file/d/114H4D0hcOjxDj_MHQNvwDBUJV3chlpdC/view?usp=sharing

Countdown of HP 67 "seconds"

Random Numbers

Snell's Law

Circle: Area and Circumference

Sphere: Surface Area and Volume

Angle Between 2 Lines with Slopes x and y

Sum of Powers

Adding Complex Numbers

Multiplying Complex Numbers

Complex Number to a Real Power

Permutation

Combination (with duplicating X and Y stack values)

Speed of Sound Approximation (in meters per second)

Finance: Present Value Annuity Factor (including setting N and I% with monthly payments)

Distance Between Two Points (x,y) and (z,t)

Horizontal Curve

Volume 2:

https://drive.google.com/file/d/1RPK2C879JeiyReox1SOTeAOQc_7QYdUN/view?usp=sharing

Sigmoid Function

Logit Function

Solving Linear Equations

Solving Monic Quadratic Polynomials (real roots only)

Intensity of a Spherical Light Source

Determinant of a 2 x 2 Matrix

Air Density of Dry Air

Time Dilation Factor

Simple Interest

Calculus: Area Under the Curve of a Linear Function

Horizontal Curve Solution given Radius and Central Angle

Freezing Altitude Levels: Dry and Wet

HP 67 Solvers:

Temperature Conversions

Cost-Sell-Margin

Decibel Gain/Loss

Eddie

All original content copyright, © 2011-2025. 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.

Casio fx-991 CW: Solving Linear Systems of Complex Numbers

Casio fx-991 CW: Solving Linear Systems of Complex Numbers

Introduction

The fx-991 CW is a capable calculator. It even handles complex number calculations and linear systems up to four variables. So we should be able to handle problems such as:

(A + Bi) * (x + yi) + (C + Di) * (z + ti) = E + Fi

(G +Hi) * (x + yi) + (J + Ki) * (z + ti) = L + Mi

where i=√-1 and A, B, C, D, E, F, G, H, J, K, L, and M are real and imaginary parts of complex numbers.

Well, yes. But the Equation mode does not allow for complex numbers, and the complex mode handles arithmetic and rectangular/polar conversions. True. However, all is not lost.

If we multiply the left side of both equations, we get:

(A * x – B * y + C * z – D * t) + (B * x + A * y + D * z + C * t)i = E + Fi

(G * x – H * y + J * z – K * t) + (H * x + G * y + K * z + J * t)i = L + Mi

Equating the real and imaginary parts, we get four equations since the real and imaginary parts are separated by addition.

A * x – B * y + C * z – D * t = E (real part)

B * x + A * y + D * z + C * t = F (imaginary part)

G * x – H * y + J * z – K * t = L (real part)

H * x + G * y + K * z + J * t = M (imaginary part)

In this form, we can use the linear system solver of the fx-991CW (as well as many other calculators).

In matrix form:

[ [ A, -B, C, D ] [ [ x ]   [ [ E ]

[ B, A, D, C ]   * [ y ]  = [ F ]

[ G, -H, J, -K ]   [ z ]    [ L ]

[ H, G, K, J ] ]   [ t ] ]  [ M ] ]

Casio fx-991CW: Algorithm

Step 1. Press the [ HOME ] key and select Equation.

Step 2. Select Simul Equation.

Step 3. Select 4 Unknowns.

Step 4. Set up the equations as follows:

A * x – B * y + C * z – D * t = E

B * x + A * y + D * z + C * t = F

G * x – H * y + J * z – K * t = L

H * x + G * y + K * z + J * t = M

Step 5. Press [ EXE ], or [ SHIFT ] [ EXE ] (≈) for all approximate solutions. Scroll to see the results x, y, z, and t. In the default setting, every answer, except for x is expressed in standard form. I’m not sure why this is. This is your solution to x + yi and z + ti.

Step 6. To do a new calculation, press [ EXE ] again.

Examples

Example 1:

(6 + 4i) * (x + yi) + (2 + 5i) * (z + ti) = 11 + 63i

(8 + 14i) * (x + yi) + (16 + 7i) * (z + ti) = 213 + 105i

Set up the four equations as:

6x – 4y + 2z – 5t = 11

4x + 6y + 5z + 2t = 63

8x -14y + 16z – 7t = 213

14x + 8y + 7z + 16t = 105

Results:

x ≈ -1.682508574

y = -2109 / 2041 ≈ -1.033317001

z = 2239 / 157 ≈ 14.2611465

t = 363 / 157 ≈ 2.312101911

x + yi ≈ -1.682508574 – 1.033317001i

z + ti ≈ 14.2611465 + 2.312101911i

Example 2:

(3 + 3i) * (x + yi) + (-9 + 7i) * (z + ti) = -4 + 2i

(-5 + 3i) * (x + yi) +(2 + 8i) * (z + ti) = 11 + 0i

Set up the four equations as:

3x – 3y - 9z – 7t = -4

3x + 3y + 7z - 9t = 2

-5x - 3y + 2z – 8t = 11

3x - 5y + 8z + 2t = 0

Results:

x ≈ -1.252444271

y = -1245 / 5114 ≈ -0.2434493547

z = 2131 / 5114 ≈ 0.4166992569

t = -2029 / 5114 ≈ -0.3967540086

x + yi ≈ -1.252444271 – 0.2434493547i

z + ti ≈ 0.4166992569 – 0.3967540086i

Until next time, take care,

Eddie

All original content copyright, © 2011-2025. 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.

RPN with HP 15C & DM32: Solving Simple Systems

RPN with HP 15C & DM32: Solving Simple Systems

Welcome to another edition of RPN with HP 15C & DM32.

Many Approaches to a Solving Problems

This blog covers two ways to solve the simple linear system of two equations:

x + y = a

x – y = b

In matrix form:

[ [ 1, 1 ] [ 1, - 1] ] * [ [ x ] [ y ] ] = [ [ a ] [ b ]

The inverse of the coefficient matrix [ [ 1, 1 ] [ 1, - 1] ]:

[ [ 1, 1 ] [ 1, - 1] ] ^-1 = [ [ 0.5, 0.5 ] [ -0.5, 0.5 ] ]

[ [ x ] [ y ] ] = [ [ 0.5, 0.5 ] [ -0.5, 0.5 ] ] * [ [ a ] [ b ] ]

The solution to the system is:

x = (a + b) / 2

y = (a – b) / 2

As far as keying the solutions in the calculator, we can address this two ways. The first, and probably the easier method, is to use memory registers. The second is to use stack operations such as swap (x<>y), roll up (R↑), roll down (R↓), enter (ENTER) as duplication, and the Last X function (LST x). Both the HP 15C and DM32 use a four-level stack.

The Memory Register Approach

Start with the stack as follows:

Y: a

X: b

The results are presented as:

Y: y

X: x

Memory Registers used:

HP 15C: R1 = a, R2 = b

DM32: A, B

HP 15C Code:

001	42, 21, 11	LBL A
002	44, 2	STO 2
003	34	X<>Y
004	44, 1	STO 1
005	30	-
006	16	CHS
007	2	2
008	10	÷
009	45, 2	RCL 2
010	45, 40, 1	RCL + 1
011	2	2
012	10	÷
013	43, 32	RTN

DM32 Code (31 bytes):

T01 LBL T

T02 STO B

T03 x<>y

T04 STO A

T05 -

T06 +/-

T07 2

T08 ÷

T09 RCL B

T10 RCL + A

T11 2

T12 ÷

T13 RTN

The Stack Approach

Start with the stack as follows:

Y: a

X: b

The results are presented as:

T : original contents of the z stack

Z: original contents of the z stack

Y: y

X: x

This time no memory registers are used

HP 15C Code:

001	42, 21, 12	LBL B
002	40	+
003	43, 36	LST x
004	2	2
005	20	×
006	34	X<>Y
007	36	ENTER
008	33	R↓
009	30	-
010	16	CHS
011	2	2
012	10	÷
013	43, 33	R↑
014	2	2
015	10	÷
016	43, 32	RTN

DM32 Code (24 bytes):

S01 LBL S

S02 +

S03 LAST x

S04 2

S05 ×

S06 x<>y

S07 ENTER

S08 R↓

S09 -

S10 +/-

S11 2

S12 ÷

S13 R↑

S14 2

S15 ÷

S16 RTN

Examples

Example 1:

x + y = 32

x – y = 16

32 ENTER 16 (run program)

Results: y = 8, x = 24

Example 2:

x + y = 9

x – y = -8

Results: y = 8.5, x = 0.5

Until next time,

Eddie

All original content copyright, © 2011-2025. 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.