Mathematics · Trigonometry
Bearing Components Calculator
Resolve a distance along a compass bearing into north and east components.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- North=10cos90°=6.123233995736766e-16.
- East=10sin90°=10.
Understand Bearing components
One idea, three depths
Choose how deeply to explain Bearing components
Bearing components: Resolve a distance along a compass bearing into north and east components.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Bearing components to answer this question: resolve a distance along a compass bearing into north and east components? Enter Distance d and Bearing θ; the calculator shows North component. For example: 10 units at 90° gives east 10 and north 0. The answer tells you North component.
Age 15Explain it to a 15-year-oldConnect it to the formula
Compass bearings are measured clockwise from north, changing the usual Cartesian component order. The rule is north=d cosθ; east=d sinθ. Its input values are Distance d, Bearing θ (°), and the main result is North component. For example: 10 units at 90° gives east 10 and north 0.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated bearing components relation over the valid real-number domain stated below. The implemented relation is north=d cosθ; east=d sinθ, evaluated from Distance d, Bearing θ (°) to produce North component. Compass bearings are measured clockwise from north, changing the usual Cartesian component order. A bearing is measured from north, not from the positive x-axis.
Inputs and valid domain
- Distance d must be a finite real number, at least 0.
- Bearing θ must be a finite real number in °.
Important boundary: A bearing is measured from north, not from the positive x-axis.
The formula
north=d cosθ; east=d sinθ
How the calculator works through it
It substitutes Distance d, Bearing θ into the formula and exposes every numerical step above. The main output is North component, accompanied by East component.
Read the result correctly
The North component is the direct answer to “resolve a distance along a compass bearing into north and east components.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
10 units at 90° gives east 10 and north 0.
Where this model stops being reliable
A bearing is measured from north, not from the positive x-axis.
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 Bearing components works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Bearing components uses north=d cosθ; east=d sinθ. 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 Bearing components.
Review this foundation about 5 min
Optional enrichment
- Functions and their graphs
Function graphs show how the Bearing components 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 Distance d, Bearing θ.
- Evaluate the principal relationship: north=d cosθ; east=d sinθ.
- Return North component and check the domain conditions described above.
Python
from math import *
def bearing_components(a, b) -> float:
return (a * cos(((b * pi) / 180.0)))
assert abs(bearing_components(10, 90) - 6.123233995736766e-16) < 1e-6 * max(1.0, abs(6.123233995736766e-16))
C
#include <assert.h>
#include <math.h>
double bearing_components(double a, double b) {
return (a * cos(((b * 3.141592653589793) / 180.0)));
}
int main(void) {
const double expected = 6.123233995736766e-16;
const double actual = bearing_components(10, 90);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double bearing_components(double a, double b) {
return (a * std::cos(((b * std::numbers::pi) / 180.0)));
}
int main() {
constexpr double expected = 6.123233995736766e-16;
const double actual = bearing_components(10, 90);
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 bearing_components(double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern cos
global bearing_components
section .text
bearing_components:
push rbp
mov rbp, rsp
sub rsp, 64
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-56], xmm0
movsd xmm0, [rbp-16]
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 cos wrt ..plt
movsd [rbp-32], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-32]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = bearing_components(a, b)
result = (a * cos(((b * pi) / 180.0)));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_] := (a * Cos[((b * 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). Bearing Components Calculator. MW SysArc Tools. https://math.mwsysarc.com/trigonometry/bearing-components
MLA 9
MW SysArc. “Bearing Components Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/trigonometry/bearing-components. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Bearing Components Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/trigonometry/bearing-components.
Harvard
MW SysArc (2026) ‘Bearing Components Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/trigonometry/bearing-components (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_bearing_components_2026,
author = {{MW SysArc}},
title = {Bearing Components Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/trigonometry/bearing-components},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Bearing Components Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/trigonometry/bearing-components
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Bearing components do?
Resolve a distance along a compass bearing into north and east components.
How does the Bearing components work?
The calculator applies north=d cosθ; east=d sinθ. Compass bearings are measured clockwise from north, changing the usual Cartesian component order.
What can I learn from the Bearing components?
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 .