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Euler Formula Calculator

Work out euler formula instantly with clear inputs, formula shown and shareable results.

About Euler Formula Calculator

Euler's formula states e^(iθ) = cos θ + i sin θ, connecting exponentials to rotation. Setting θ = π gives Euler's identity e^(iπ) + 1 = 0, tying together e, i, π, 1 and 0.

Formulas

Euler's formula

e^(iθ) = cos θ + i·sin θ

Euler's identity

e^(iπ) + 1 = 0

Worked example

With the default values below, Euler Formula Calculator returns r·e^(iθ) in rectangular form of -1 + 0i. Change any input above and every figure here is recalculated in your browser.

Inputs used

Angle θ
180 °
Modulus rminimum 0
1

Results

r·e^(iθ) in rectangular form
-1 + 0i
r·cos θ
-1.00000000
r·sin θ
0.00000000
Angle in radians
3.14159265

Step by step

  1. 1Values used
  2. 2Euler's formula
  3. 3Euler's identity
  4. 4r·e^(iθ) in rectangular form= -1 + 0i
  5. 5r·cos θ= -1.00000000
  6. 6r·sin θ= 0.00000000
  7. 7Angle in radians= 3.14159265

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is e^(iπ)?

cos 180° + i sin 180° = −1 exactly, which is Euler's identity.

Why does an exponential produce rotation?

The power series for e^(iθ) splits into the cosine and sine series, so multiplying by i repeatedly cycles through the four directions.

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