Mathematics · Trigonometry
Pythagorean Trigonometric Identity Calculator
Evaluate sine, cosine and the identity sin²θ+cos²θ=1 at any angle.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Convert 37° to 0.6457718232379019 radians.
- Square the values: 0.6018150231520483²+0.7986355100472928²=1.
- The result equals 1 within floating-point precision.
Understand Trig identity
One idea, three depths
Choose how deeply to explain Trig identity
Trig identity: Evaluate sine, cosine and the identity sin²θ+cos²θ=1 at any angle.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Trig identity to answer this question: evaluate sine, cosine and the identity sin²θ+cos²θ=1 at any angle? Enter Angle θ; the calculator shows sin²θ + cos²θ. For example: At 37°, sin²(37°)+cos²(37°)=1 apart from floating-point rounding. The answer tells you sin²θ + cos²θ.
Age 15Explain it to a 15-year-oldConnect it to the formula
The identity follows from x²+y²=1 on the unit circle and is fundamental when simplifying wave and derivative expressions. The rule is sin²θ+cos²θ=1. Its input values are Angle θ (°), and the main result is sin²θ + cos²θ. For example: At 37°, sin²(37°)+cos²(37°)=1 apart from floating-point rounding.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated trig identity relation over the valid real-number domain stated below. The implemented relation is sin²θ+cos²θ=1, evaluated from Angle θ (°) to produce sin²θ + cos²θ. The identity follows from x²+y²=1 on the unit circle and is fundamental when simplifying wave and derivative expressions. Square the function values, not the angle: (sin θ)² is not sin(θ²).
Inputs and valid domain
- Angle θ must be a finite real number in °.
Important boundary: Square the function values, not the angle: (sin θ)² is not sin(θ²).
The formula
sin²θ+cos²θ=1
How the calculator works through it
It substitutes Angle θ into the formula and exposes every numerical step above. The main output is sin²θ + cos²θ, accompanied by sin θ, cos θ, Rounding error from 1.
Read the result correctly
The sin²θ + cos²θ is the direct answer to “evaluate sine, cosine and the identity sin²θ+cos²θ=1 at any angle.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
At 37°, sin²(37°)+cos²(37°)=1 apart from floating-point rounding.
Where this model stops being reliable
Square the function values, not the angle: (sin θ)² is not sin(θ²).
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 Trig identity works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Trig identity uses sin²θ+cos²θ=1. 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 Trig identity.
Review this foundation about 5 min
Optional enrichment
- Functions and their graphs
Function graphs show how the Trig identity 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 Angle θ.
- Evaluate the principal relationship: sin²θ+cos²θ=1.
- Return sin²θ + cos²θ and check the domain conditions described above.
Python
from math import *
def pythagorean_trig_identity(x) -> float:
return ((sin(((x * pi) / 180.0)) * sin(((x * pi) / 180.0))) + (cos(((x * pi) / 180.0)) * cos(((x * pi) / 180.0))))
assert abs(pythagorean_trig_identity(37) - 1) < 1e-6 * max(1.0, abs(1))
C
#include <assert.h>
#include <math.h>
double pythagorean_trig_identity(double x) {
return ((sin(((x * 3.141592653589793) / 180.0)) * sin(((x * 3.141592653589793) / 180.0))) + (cos(((x * 3.141592653589793) / 180.0)) * cos(((x * 3.141592653589793) / 180.0))));
}
int main(void) {
const double expected = 1;
const double actual = pythagorean_trig_identity(37);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double pythagorean_trig_identity(double x) {
return ((std::sin(((x * std::numbers::pi) / 180.0)) * std::sin(((x * std::numbers::pi) / 180.0))) + (std::cos(((x * std::numbers::pi) / 180.0)) * std::cos(((x * std::numbers::pi) / 180.0))));
}
int main() {
constexpr double expected = 1;
const double actual = pythagorean_trig_identity(37);
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 pythagorean_trig_identity(double x)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern sin
extern cos
global pythagorean_trig_identity
section .text
pythagorean_trig_identity:
push rbp
mov rbp, rsp
sub rsp, 192
movsd [rbp-8], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-56], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-56]
movsd [rbp-48], xmm0
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-64], xmm0
movsd xmm0, [rbp-48]
divsd xmm0, [rbp-64]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-40]
call sin wrt ..plt
movsd [rbp-32], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-96], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-96]
movsd [rbp-88], xmm0
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-104], xmm0
movsd xmm0, [rbp-88]
divsd xmm0, [rbp-104]
movsd [rbp-80], xmm0
movsd xmm0, [rbp-80]
call sin wrt ..plt
movsd [rbp-72], xmm0
movsd xmm0, [rbp-32]
mulsd xmm0, [rbp-72]
movsd [rbp-24], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-144], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-144]
movsd [rbp-136], xmm0
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-152], xmm0
movsd xmm0, [rbp-136]
divsd xmm0, [rbp-152]
movsd [rbp-128], xmm0
movsd xmm0, [rbp-128]
call cos wrt ..plt
movsd [rbp-120], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-184], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-184]
movsd [rbp-176], xmm0
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-192], xmm0
movsd xmm0, [rbp-176]
divsd xmm0, [rbp-192]
movsd [rbp-168], xmm0
movsd xmm0, [rbp-168]
call cos wrt ..plt
movsd [rbp-160], xmm0
movsd xmm0, [rbp-120]
mulsd xmm0, [rbp-160]
movsd [rbp-112], xmm0
movsd xmm0, [rbp-24]
addsd xmm0, [rbp-112]
movsd [rbp-16], xmm0
movsd xmm0, [rbp-16]
leave
ret
MATLAB
function result = pythagorean_trig_identity(x)
result = ((sin(((x * pi) / 180.0)) * sin(((x * pi) / 180.0))) + (cos(((x * pi) / 180.0)) * cos(((x * pi) / 180.0))));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[x_] := ((Sin[((x * Pi) / 180.0)] * Sin[((x * Pi) / 180.0)]) + (Cos[((x * Pi) / 180.0)] * Cos[((x * Pi) / 180.0)]));
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). Pythagorean Trigonometric Identity Calculator. MW SysArc Tools. https://math.mwsysarc.com/trigonometry/pythagorean-identity
MLA 9
MW SysArc. “Pythagorean Trigonometric Identity Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/trigonometry/pythagorean-identity. Accessed 4 Sept. 2026.
Chicago 17
MW SysArc. “Pythagorean Trigonometric Identity Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed September 4, 2026. https://math.mwsysarc.com/trigonometry/pythagorean-identity.
Harvard
MW SysArc (2026) ‘Pythagorean Trigonometric Identity Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/trigonometry/pythagorean-identity (Accessed: 4 September 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_pythagorean_trig_identity_2026,
author = {{MW SysArc}},
title = {Pythagorean Trigonometric Identity Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/trigonometry/pythagorean-identity},
note = {Published July 21, 2026; accessed September 4, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Pythagorean Trigonometric Identity Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-09-04
UR - https://math.mwsysarc.com/trigonometry/pythagorean-identity
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Trig identity do?
Evaluate sine, cosine and the identity sin²θ+cos²θ=1 at any angle.
How does the Trig identity work?
The calculator applies sin²θ+cos²θ=1. The identity follows from x²+y²=1 on the unit circle and is fundamental when simplifying wave and derivative expressions.
What can I learn from the Trig identity?
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 .