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. 2nd 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
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