If z = 2(cos(θ) + i sin(θ)), what is the value of z when θ = π/3?

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Question: If z = 2(cos(θ) + i sin(θ)), what is the value of z when θ = π/3?

Options:

Correct Answer: 1 + √3i

Solution:

z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + √3i.

If z = 2(cos(θ) + i sin(θ)), what is the value of z when θ = π/3?

If z = 2(cos(θ) + i sin(θ)), what is the value of z when θ = π/3?

Step 1: Identify the given equation: z = 2(cos(θ) + i sin(θ)).
Step 2: Substitute θ with π/3 in the equation: z = 2(cos(π/3) + i sin(π/3)).
Step 3: Find the value of cos(π/3). The value is 1/2.
Step 4: Find the value of sin(π/3). The value is √3/2.
Step 5: Substitute the values of cos(π/3) and sin(π/3) into the equation: z = 2(1/2 + i√3/2).
Step 6: Simplify the expression: z = 2 * (1/2) + 2 * (i√3/2).
Step 7: Calculate 2 * (1/2) = 1 and 2 * (i√3/2) = √3i.
Step 8: Combine the results: z = 1 + √3i.

Complex Numbers – Understanding the representation of complex numbers in polar form and converting them to rectangular form.
Trigonometric Functions – Knowledge of the values of cosine and sine for common angles, particularly π/3.
Euler's Formula – Application of Euler's formula in expressing complex numbers in terms of trigonometric functions.