Express the complex number in trigonometric form: 3 - 3i.
Complex numbers are very important concepts in mathematics. These are the numbers that can't be represented on the number line or the cartesian plane. They can have both real and imaginary parts.
Answer: The complex number 3 - 3i can be represented in trigonometric form as 3√2 (cos(−π/4) + i sin(−π/4)).
Let's understand the solution in detail.
To convert a complex number z to trigonometric form, we use the formula:
⇒ z = |z| (cos θ + i sin θ) --- (1)
Here, |z| is the magnitude of z and θ is the phase of z.
Here, we have z = 3 - 3i in the form of z = a + bi; here a = 3 and b = -3.
Now, we know that |z| = √(a2 + b2) = √(32 + 32) = 3√2.
⇒ tan θ = b/a = -3/3 = -1
⇒ θ = -π/4
Hence, after we substitute the values of |z| and θ in equation (1), we get 3√2 (cos(−π/4) + i sin(−π/4)).
This is the polar form of the complex number 3 - 3i
Hence, the complex number 3 - 3i can be represented in trigonometric form as 3√2 (cos(−π/4) + i sin(−π/4)).