Cis — Definition, Formula & Examples

Cis

A complex-valued function made from sine and cosine with definition cis θ = cos θ + isin θ.

Note: cis θ is the same as e^iθ.

Key Formula

\operatorname{cis} \theta = \cos \theta + i \sin \theta = e^{i\theta}

Where:

$\theta$ = The angle in radians (or degrees) measured from the positive real axis
$i$ = The imaginary unit, where i² = −1
$\cos \theta$ = The real part of the complex number
$\sin \theta$ = The imaginary part of the complex number

Worked Example

Problem: Express cis 60° in the form a + bi.

Step 1: Write out the definition of cis using the given angle.

\operatorname{cis} 60° = \cos 60° + i \sin 60°

Step 2: Evaluate cos 60°.

\cos 60° = \frac{1}{2}

Step 3: Evaluate sin 60°.

\sin 60° = \frac{\sqrt{3}}{2}

Step 4: Combine the results into a + bi form.

\operatorname{cis} 60° = \frac{1}{2} + \frac{\sqrt{3}}{2}\,i

Answer: cis 60° = 1/2 + (√3/2)i

Another Example

This example includes a modulus r ≠ 1, showing how cis is used in the full polar form r cis θ to convert a complex number to rectangular form.

Problem: Write the complex number z = 3 cis 150° in rectangular form a + bi.

Step 1: A complex number in polar form is written as r cis θ, where r is the modulus. Here r = 3 and θ = 150°. Expand using the definition.

z = 3(\cos 150° + i \sin 150°)

Step 2: Evaluate cos 150°. Since 150° is in the second quadrant, cosine is negative.

\cos 150° = -\frac{\sqrt{3}}{2}

Step 3: Evaluate sin 150°. Sine is positive in the second quadrant.

\sin 150° = \frac{1}{2}

Step 4: Distribute r = 3 to both the real and imaginary parts.

z = 3\left(-\frac{\sqrt{3}}{2}\right) + 3\left(\frac{1}{2}\right)i = -\frac{3\sqrt{3}}{2} + \frac{3}{2}\,i

Answer: z = −(3√3)/2 + (3/2)i

Frequently Asked Questions

What is the difference between cis θ and e^(iθ)?

They are exactly the same thing. The notation cis θ is simply a shorthand for cos θ + i sin θ, and Euler's formula tells us that e^(iθ) = cos θ + i sin θ. Some textbooks prefer cis because it avoids introducing the exponential function, while others prefer e^(iθ) because it follows the standard rules of exponents.

Why is cis notation useful?

Cis notation makes multiplying and dividing complex numbers in polar form much cleaner. When you multiply two complex numbers, you can write r₁ cis θ₁ · r₂ cis θ₂ = r₁r₂ cis(θ₁ + θ₂), which is far more compact than writing out all the cosine and sine terms. It is especially helpful when applying De Moivre's Theorem.

How do you multiply two complex numbers using cis?

To multiply r₁ cis θ₁ by r₂ cis θ₂, multiply the moduli and add the angles: the product is r₁r₂ cis(θ₁ + θ₂). For division, divide the moduli and subtract the angles: (r₁/r₂) cis(θ₁ − θ₂). This rule comes directly from the product-to-sum identities for sine and cosine.

cis θ notation vs. e^(iθ) notation

	cis θ notation	e^(iθ) notation
Definition	cos θ + i sin θ (abbreviation of cosine + i sine)	cos θ + i sin θ (from Euler's formula)
Formula	cis θ = cos θ + i sin θ	e^(iθ) = cos θ + i sin θ
Multiplication rule	cis α · cis β = cis(α + β)	e^(iα) · e^(iβ) = e^(i(α+β))
Typical context	Precalculus and introductory courses	Calculus, engineering, and physics
Requires knowledge of e	No — purely trigonometric	Yes — uses Euler's number e ≈ 2.718

Why It Matters

You encounter cis notation when working with complex numbers in polar form, particularly in precalculus and early college mathematics. It is central to De Moivre's Theorem, which lets you raise complex numbers to integer powers efficiently: (r cis θ)^n = r^n cis(nθ). Electrical engineers also rely on this notation (often written as the phasor form) to analyze alternating current circuits.

Common Mistakes

Mistake: Confusing the real and imaginary parts by writing cis θ = sin θ + i cos θ.

Correction: Remember the name: 'c-i-s' stands for cos + i sin, in that exact order. The cosine is the real part, and the sine (multiplied by i) is the imaginary part.

Mistake: Forgetting to multiply both the cosine and sine by the modulus r when converting r cis θ to rectangular form.

Correction: The modulus r scales the entire complex number: r cis θ = r cos θ + i r sin θ. Both components must be multiplied by r.

Related Terms

Complex Numbers — Numbers of the form a + bi that cis produces
Polar Form of a Complex Number — Uses r cis θ to represent complex numbers
e (Euler's Number) — Base of the equivalent expression e^(iθ)
Sine — The imaginary component in cis θ
Cosine — The real component in cis θ
Trig Functions — Sine and cosine are the building blocks of cis
Function — Cis is a function from angles to complex numbers