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.

Spotlight: AccuMath 400B Slide Rule

Spotlight: AccuMath 400B Slide Rule

Introduction

I purchased an AccuMath 400B slide rule at Antique Station, an antique store in Oro Grande just north of Victorville, CA.

I am impressed of the larger markings on the slide rule. The slide rule very easy to read, yet has a lot of scales for calculations including powers, roots, trigonometry, and logarithms.

The Scales

The slide rules have the following scales:

Top Frame (stationary):

S: Sine of angles in degrees. A = sin(S°) ÷ 100, S° = arcsin(A × 100)

K: Cube and cube root scale associated with scale D. K = D³, D = ³√K

A: Square and square root scale associated with scale D. A = D², D = √A

Slide:

B: Square and square root scale. Scale is from 1 to 100 and it is the same scale as Scale A.

CI: Reciprocal of scale C.

C: Multiplication and division scale from 1 to 10. Same as scale D.

Bottom Frame (stationary):

D: Multiplication and division scale from 1 to 10.

L: Logarithmic and exponential scale associated with D. L = log D, D = 10^L. (base 10 logs)

T: Tangent of angles in degrees. D = tan(T°) ÷ 10, T° = arctan(D × 10)

Note: A lot of slide rules associate the S scale with the C/D scales, but for this particular design, the A scale is used instead.

The Back Side

U.S./Metric Conversions, Equivalents, and Settings (i.e. 1 in mercury = 1.133 ft water)

Fractions up to 64ths and their decimal equivalents (i.e. 47/64 = 0.734375)

Trigonometric Identities, Right Triangles, Law of Sines and Cosines

Example Calculations

Keep your exponents in mind! N = mantissa * 10^exponent

Hairline: the plastic cursor you move around, Slide: the center piece of the slide rule you move around

4 × 2 = 8

Slide C right to match C: 1, D: 4

Move cursor right to C = 2

Read down to D: 8

3 × 9 = 27

Rearrange to 9 × 3. Slide C left to match C:1, D: 9

Move cursor left to C = 3

Read down to D: 2.7

Multiply by 10 since we slide C left: 2.7 × 10 = 27

12 × 33 = 396 (Approximate method)

Rearrange to 33 × 12. Slide B left to match: B: 1, A: 33

Move cursor left to B = 12

Read on A: the cursor is very close to 4.

Multiply by 100 since we moved B left. Result: approx 400.

54 ÷ 9 = 6

Move Slide B to match: A = 54, B = 9

Move cursor left to B = 1

Read A = 6

√81 = 9, 9^3 = 729 (approximate)

On A, slide cursor to 81.

Read on D: 9 (square root)

Read on K: in between 720 and 730. (so ≈725?)

Note that the scales are limited.

√144 = 12 (√(144 ÷ 100 × 100) = √1.44 × √100 = √1.44 × 10)

On A, slide cursor to 1.44.

Read on D: 1.2

Multiply by 10. Result: 12.

Logarithms: D and L scales. L = log D, D = 10^L

log 3 ≈ 0.47712

On D, slide cursor to 3.

Read on L: about 0.47

10^0.55 ≈ 3.54814

On L, slide cursor to 0.55

Read on D: about 3.54

Tangent of Angles in degrees: D and T scales. D = tan(T°) ÷ 10, T° = arctan(10 × D)

Tan(25°) ≈ 0.46631 (approximate)

On T, slide cursor to 25.

Read on D: about 4.65. Divide by 10 for a result of approximately 0.465

arctan(0.7) ≈ 34.99202 (approximate)

On D, slide to 7 (0.7 × 10)

Read on T: close to 35. Result: approximately 35°

Sine of Angles in degrees: A and S scales (for this model!) A = sin(S°)÷100, S° = arcsin(A×100)

sin(30°) = 0.5

On S, slide cursor to 30.

Read on A: 50. Divide by 100 for a final answer of 0.5

arcsin(0.2) ≈ 11.53700°

On A, slide to 20. (0.2 × 100)

Read on S: slightly after 11.5, for an approximation of 11.5°.

Reciprocal: CI and D scales. When the slide is at its home position, that is C = 1 and D = 1 are aligned:

1/CI = D

Find 1 / 4 = 0.25

Slide D to 4.

Read CI at 2.5 and divide by 10. Result: 0.25

Source

Instruction Manual for the AccuMath 400B.

Page 1: https://www.sliderule.ca/stermf.jpg

Page 2: https://www.sliderule.ca/stermb.jpg

Eric’s Slide Rule Page, last updated December 1, 2002. Accessed June 27, 2026.

Best,

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.

Basic vs. Python: Helix Curve (with Casio fx-CG 50)

Basic vs. Python: Helix Curve (with Casio fx-CG 50)

Calculators Used: Casio fx-CG100, Casio fx-CG50

The Helix Curve

The helix space curve can be defined with the following set of parametric equations:

x(t) = r × cos(t)

y(t) = r × sin(t)

z(t) = c × t

where r = radius, c = spacing between the coils of the helix

We can use any measurement of length we want, such as meters, feet, or inches, as long as our measurements are consistent.

Graphing the Helix Equation

Regarding calculators, 3D parametric equations can be graphed with the Casio fx-CG 50, fx-CG 100 (independent variables s and t), and the TI-Nspire (independent variables t and u). The screenshots below is a graph of a helix with the use of the fx-CG 100 emulator (classpad.workspace.com):

Curvature, Torsion, and Arc Length of a Helix

For the formulas, let

x = r × cos(t), x’ = -r × sin(t), x’’ = -r × cos(t), x’’’ = r × sin(t)

y = r × sin(t), y’ = r × cos(t), y’’ = -r × sin(t), y’’’ = -r × cos(t)

z = c × t, z’ = c, z’’ = 0, z’’’ = 0

The variable t is the independent variable of x(t), y(t), and z(t).

Curvature

The general formula for curvature:

k² = ((x’² + y’² + z’²) × (x’’² + y’’² + z’’²) – (x’ × x’’ + y’ × y’’ + z’ × z’’)) ÷ (x’² + y’² + z’²)³

Applying to the helix:

k² = ((r² sin² t + r² cos² t + c²) × (r² cos² t + r² sin² t + 0) – (r² sin t cos t – r² sin t cos t + 0)) ÷ (r² sin² t + r² cos² t + c²)³

Note: r² sin² t + r² cos² t = r² × (sin² t + cos² t) = r²

k² = ((r² + c²) × r²) ÷ (r² + c²)³

k² = r² ÷ (r² + c²)²

k = r ÷ (r² + c²)

Note: curvature is assumed to be a positive value

Torsion

The general formula for torsion:

τ =

(x’’’ × (y’ × z’’ – y’’ × z’) + y’’’ × (x’’’ × z’ – x’ × z’’’) + z’’’ × (x’ × y’’ – x’’ × y’))

÷ ((y’ × z’’ – y’’ × z’)² + (x’’ × z’ – x’ × z’’)² + (x’ × y’’ – x’’ × y’)²)

Breaking it down into parts:

x’’’ × (y’ × z’’ – y’’ × z’) = r × sin t × (0 - -r × sin t × c) = r² × c × sin² t

y’’’ × (x’’’ × z’ – x’ × z’’’) = -r × cos t × (-r × cos t × c – 0) = r² × c × cos² t

z’’’ × (x’ × y’’ – x’’ × y’) = 0

x’’’ × (y’ × z’’ – y’’ × z’) + y’’’ × (x’’’ × z’ – x’ × z’’’) + z’’’ × (x’ × y’’ – x’’ × y’)

= r² × c × sin² t + r² × c × cos² t + 0 = r² × c

(y’ × z’’ – y’’ × z’)² = (0 - -r × sin t × c)² = r² × c² × sin² t

(x’’ × z’ – x’ × z’’)² = (-r × cos t × c – 0)² = r² × c² × cos² t

(x’ × y’’ – x’’ × y’)² = (r² sin² t + r² cos² t)² = r^4

(y’ × z’’ – y’’ × z’)² + (x’’ × z’ – x’ × z’’)² + (x’ × y’’ – x’’ × y’)²

= r² × c² × sin² t + r² × c² × cos² t + r^4 = r² × c² + r^4 = r² × (c² + r²)

Then:

τ = (r² × c) ÷ (r² × (c² + r²)) = c ÷ (r² + c²)

Arc Tangent from t = 0 to t = x

s = ∫ √(x’² + y’² + z’²) dt from t = 0 to t = x

Since:

x’² + y’² + z’² = r² sin² t + r² cos² t + c² = r² + c²

Then:

s = ∫ √(x’² + y’² + z’²) dt from t = 0 to t = x

= ∫ √(r² + c²) dt from t = 0 to t = x

= x × √(r² + c²)

To summarize, for the helix curve:

Curvature: k = r ÷ (r² + c²)

Torsion: τ = c ÷ (r² + c²)

Arc Length to x: s = x × √(r² + c²)

The code below calculates the following:

* curvature

* torsion

* arc length to 2π

Casio fx-CG50 Program HELIXFX

"HELIX: CASIO BASIC"

"RADIUS"?→R

"SPACING"?→c

"CURVATURE="

R÷(R²+C²)→K ◢

"TORISON="

C÷(R²+C²)→T ◢

"ARC LENGTH TO 2π="

2×π×√(R²+C²)→S

Python Script: helixp.py

from math import *

print("Helix: Parameters")

print("math module imported\n")

r=eval(input("radius? "))

c=eval(input("spacing? "))

k=r/(r**2+c**2)

t=c/(r**2+c**2)

print("curvature=\n",str(k))

print("torsion=\n",str(t))

print("arc length to 2pi=\n",str(s))

Example

Radius: r = 2.75

Spacing: c = 0.89

Outputs:

Curvature: 0.3291599837

Torsion: 0.1065281402

Arc length to 2π: 18.16112298

Sources

Harris, John W. and Horst Stocker. Handbook of Mathematics and Computational Science Springer: New York, NY. 2006. ISBN 978-0-387-94746-4

Lee, Sarah. “Curvature and Torsion of 3D Parametric Curves.” Number Analytics // Super Easy Data Analysis Tool for Research, May 17, 2015, www.numberanalytics.com/blog/curvature-torsion-3d-parametric-curves . Accessed 02 July 2025.

Weisstein, Eric W. "Helix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Helix.html Accessed July 2, 2025.

Wikimedia Foundation. “Torsion of a curve.” Wikipedia. Lasted Edited January 2, 2023, https://en.wikipedia.org/wiki/Torsion_of_a_curve Accessed July 2, 2025.

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.

The author does not use AI engines and never will.