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 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: At 37°, sin²(37°)+cos²(37°)=1 apart from floating-point rounding. The answer tells you sin²θ + cos²θ.
Age 15 Explain it to a 15-year-old Connect 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.
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 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. Ambiguous triangle data, rounded angles and the wrong degree/radian mode can produce a plausible-looking but incorrect result. 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 value 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
Ambiguous triangle data, rounded angles and the wrong degree/radian mode can produce a plausible-looking but incorrect result. In particular, 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
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 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.
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
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.