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.
