A complex number z = a + bi can be pictured as a point or vector in the complex plane, where a is the horizontal real part and b is the vertical imaginary part. The trigonometric form describes the same number using its distance from the origin and its direction angle. This form is useful because it makes multiplication, division, powers, and roots much easier to understand and calculate.
It connects algebra with geometry by showing every nonzero complex number as a rotation and a scaling.
Understanding Math: Trigonometric Form of Complex Numbers
To convert a number from rectangular form, first find its length from the origin. Then find an angle that points into the correct quadrant. This second step needs care.
An inverse tangent gives a reference angle, but it may not identify the correct direction by itself. A point with a negative real part needs an angle on the left side of the plane. A point below the horizontal axis needs a negative angle or an equivalent positive angle.
Calculator functions often include atan2, which uses both coordinate signs and handles this issue well. Keep angle units consistent.
If an angle is in degrees, every trigonometric calculation must use degrees. If it is in radians, stay in radians.
The rule for multiplication follows from the angle addition formulas for sine and cosine. When two complex numbers are multiplied, their lengths multiply while their directions combine by adding angles. This has a clear geometric meaning.
A factor with length greater than one stretches a vector. A factor with length between zero and one shrinks it. A factor whose length is one changes only direction.
The imaginary unit is a useful example because multiplying by it turns a vector by one quarter turn counterclockwise. Repeated multiplication therefore creates repeated rotations.
Division reverses this process. The length is divided, and the divisor angle is subtracted.
De Moivre's theorem makes repeated multiplication manageable. Raising a complex number to a whole number power raises its length to that power and multiplies its angle by that whole number. This explains patterns that look difficult in rectangular form.
A number with length one returns to its starting position after enough equal rotations when its angle is a suitable fraction of a full turn. Roots work in the opposite direction. To find an nth root, take an nth root of the length and divide the angle by n.
However, the original direction can include any whole number of full turns. These choices produce n different roots for every nonzero complex number. Their points lie evenly spaced around a circle.
Complex trigonometric form appears whenever size and phase matter. In electrical circuits with alternating current, voltage and current can have the same frequency but different phase angles. Engineers use complex numbers to combine these quantities without tracking sine waves at every instant.
Similar ideas appear in sound processing, radio signals, computer graphics, and rotations in physics. In school problems, pay attention to the zero case because zero has length zero but no unique direction. Check quadrant choices before simplifying powers or roots.
Finally, remember that angles differing by a full turn represent the same nonzero complex number. This fact is essential for roots, since missing extra full turns means missing valid answers.
Key Facts
- Rectangular form: z = a + bi, where a is the real part and b is the imaginary part.
- Magnitude: r = |z| = sqrt(a^2 + b^2).
- Argument: θ satisfies cos θ = a/r and sin θ = b/r, with the quadrant determined by a and b.
- Trigonometric form: z = r(cos θ + i sin θ), often written z = rcis θ.
- Multiplication: r1cis θ1 times r2cis θ2 = r1r2cis(θ1 + θ2).
- De Moivre's theorem: [r(cos θ + i sin θ)]^n = r^n(cos nθ + i sin nθ) for integer n.
Vocabulary
- Complex plane
- A coordinate plane where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers.
- Modulus
- The modulus of a complex number is its distance from the origin, written |z| or r.
- Argument
- The argument of a nonzero complex number is the angle its vector makes with the positive real axis.
- Trigonometric form
- Trigonometric form writes a complex number as r(cos θ + i sin θ), using magnitude and angle.
- De Moivre's theorem
- De Moivre's theorem gives a shortcut for raising a complex number in trigonometric form to an integer power.
Common Mistakes to Avoid
- Using θ = arctan(b/a) without checking the quadrant is wrong because arctangent alone may give an angle pointing to the wrong part of the complex plane.
- Forgetting that r is always nonnegative is wrong because r represents distance from the origin, not a signed coordinate.
- Adding magnitudes when multiplying complex numbers is wrong because multiplication multiplies the magnitudes and adds the angles.
- Using degrees in one step and radians in another is wrong because trigonometric functions must use a consistent angle unit throughout the calculation.
Practice Questions
- 1 Convert z = 3 + 4i to trigonometric form. Find r and an angle θ in degrees to the nearest tenth.
- 2 Let z1 = 2(cos 30° + i sin 30°) and z2 = 5(cos 80° + i sin 80°). Find z1z2 in trigonometric form.
- 3 Explain why multiplying a complex number by cos 90° + i sin 90° rotates its vector but does not change its length.