Mathematics · Trigonometry
Inverse Trigonometric Functions Calculator
Calculate arcsine, arccosine and arctangent for a real ratio and show their principal angles.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- arcsin(0.5)=30.000000000000004°.
- arccos(0.5)=60.00000000000001°.
- arctan(0.5)=26.56505117707799°.
Understand Inverse trig functions
One idea, three depths
Choose how deeply to explain Inverse trig functions
Inverse trig functions: Calculate arcsine, arccosine and arctangent for a real ratio and show their principal angles.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Inverse trig functions to answer this question: calculate arcsine, arccosine and arctangent for a real ratio and show their principal angles? Enter Ratio r; the calculator shows arcsin(r). For example: For r=0.5, asin(r)=30° and acos(r)=60°. The answer tells you arcsin(r).
Age 15Explain it to a 15-year-oldConnect it to the formula
Inverse trigonometric functions recover a principal angle from a ratio; restricted output ranges make each inverse single-valued. The rule is θ=asin(r), acos(r), or atan(r). Its input values are Ratio r, and the main result is arcsin(r). For example: For r=0.5, asin(r)=30° and acos(r)=60°.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated inverse trig functions relation over the valid real-number domain stated below. The implemented relation is θ=asin(r), acos(r), or atan(r), evaluated from Ratio r to produce arcsin(r). Inverse trigonometric functions recover a principal angle from a ratio; restricted output ranges make each inverse single-valued. Arcsine and arccosine require a ratio from −1 to 1, and each inverse returns only its principal angle.
Inputs and valid domain
- Ratio r must be a finite real number, at least -1, at most 1.
Important boundary: Arcsine and arccosine require a ratio from −1 to 1, and each inverse returns only its principal angle.
The formula
θ=asin(r), acos(r), or atan(r)
How the calculator works through it
It substitutes Ratio r into the formula and exposes every numerical step above. The main output is arcsin(r), accompanied by arccos(r), arctan(r).
Read the result correctly
The arcsin(r) is the direct answer to “calculate arcsine, arccosine and arctangent for a real ratio and show their principal angles.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
For r=0.5, asin(r)=30° and acos(r)=60°.
Where this model stops being reliable
Arcsine and arccosine require a ratio from −1 to 1, and each inverse returns only its principal angle.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Inverse trig functions works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Inverse trig functions uses θ=asin(r), acos(r), or atan(r). You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Angles in degrees and radians
Interpreting the angle convention is essential for understanding the inputs and output of Inverse trig functions.
Review this foundation about 5 min
Optional enrichment
- Functions and their graphs
Function graphs show how the Inverse trig functions relationship changes across a full angle or period.
Review this foundation about 6 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Ratio r.
- Evaluate the principal relationship: θ=asin(r), acos(r), or atan(r).
- Return arcsin(r) and check the domain conditions described above.
Python
from math import *
def inverse_trig_functions(a) -> float:
return ((180.0 / pi) * asin(a))
assert abs(inverse_trig_functions(0.5) - 30.000000000000004) < 1e-6 * max(1.0, abs(30.000000000000004))
C
#include <assert.h>
#include <math.h>
double inverse_trig_functions(double a) {
return ((180.0 / 3.141592653589793) * asin(a));
}
int main(void) {
const double expected = 30.000000000000004;
const double actual = inverse_trig_functions(0.5);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double inverse_trig_functions(double a) {
return ((180.0 / std::numbers::pi) * std::asin(a));
}
int main() {
constexpr double expected = 30.000000000000004;
const double actual = inverse_trig_functions(0.5);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double inverse_trig_functions(double a)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern asin
global inverse_trig_functions
section .text
inverse_trig_functions:
push rbp
mov rbp, rsp
sub rsp, 48
movsd [rbp-8], xmm0
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-32], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-40], xmm0
movsd xmm0, [rbp-32]
divsd xmm0, [rbp-40]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-8]
call asin wrt ..plt
movsd [rbp-48], xmm0
movsd xmm0, [rbp-24]
mulsd xmm0, [rbp-48]
movsd [rbp-16], xmm0
movsd xmm0, [rbp-16]
leave
ret
MATLAB
function result = inverse_trig_functions(a)
result = ((180.0 / pi) * asin(a));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_] := ((180.0 / Pi) * ArcSin[a]);
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Algebra and Trigonometry 2e
Read the related free OpenStax mathematics chaptersCite this book
- APA 7
- Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- MLA 9
- Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
- Chicago author-date
- Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Inverse Trigonometric Functions Calculator. MW SysArc Tools. https://math.mwsysarc.com/trigonometry/inverse-trigonometric-functions
MLA 9
MW SysArc. “Inverse Trigonometric Functions Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/trigonometry/inverse-trigonometric-functions. Accessed 4 Sept. 2026.
Chicago 17
MW SysArc. “Inverse Trigonometric Functions Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed September 4, 2026. https://math.mwsysarc.com/trigonometry/inverse-trigonometric-functions.
Harvard
MW SysArc (2026) ‘Inverse Trigonometric Functions Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/trigonometry/inverse-trigonometric-functions (Accessed: 4 September 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_inverse_trig_functions_2026,
author = {{MW SysArc}},
title = {Inverse Trigonometric Functions Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/trigonometry/inverse-trigonometric-functions},
note = {Published July 21, 2026; accessed September 4, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Inverse Trigonometric Functions Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-09-04
UR - https://math.mwsysarc.com/trigonometry/inverse-trigonometric-functions
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Inverse trig functions do?
Calculate arcsine, arccosine and arctangent for a real ratio and show their principal angles.
How does the Inverse trig functions work?
The calculator applies θ=asin(r), acos(r), or atan(r). Inverse trigonometric functions recover a principal angle from a ratio; restricted output ranges make each inverse single-valued.
What can I learn from the Inverse trig functions?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .