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 5 Explain it to a 5-year-old Start with a picture
Think of a triangle as a ramp: if you know some sides or turns, this calculator helps find the missing part. For example: For r=0.5, asin(r)=30° and acos(r)=60°. The answer tells you arcsin(r).
Age 15 Explain it to a 15-year-old Connect 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°.
College Explain it at college level State the model precisely
This calculator evaluates a trigonometry model over the stated real-valued domain. 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. Ambiguous triangle data, rounded angles and the wrong degree/radian mode can produce a plausible-looking but incorrect result. 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 at least -1 and 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
Ambiguous triangle data, rounded angles and the wrong degree/radian mode can produce a plausible-looking but incorrect result. In particular, 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
Continue with a free textbook
OpenStax reading and academic references
Use the calculator as the worked interaction, then continue into the peer-reviewed textbook context. MW SysArc links to OpenStax; the explanation on this page is original and does not reproduce the book.
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 books are free to read online. Their current reuse licence is CC BY-NC-SA; follow the licence shown on the linked book before redistributing or adapting its content.
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.
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
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.
Last reviewed 2026-07-21. Calculations tested 2026-07-21.