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.
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.