If z = 5(cos(θ) + i sin(θ)), what is the value of z when θ = π/3? (2023)

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Question: If z = 5(cos(θ) + i sin(θ)), what is the value of z when θ = π/3? (2023)

Options:

Correct Answer: 5 + 2.5i

Exam Year: 2023

Solution:

z = 5(cos(π/3) + i sin(π/3)) = 5(1/2 + i√3/2) = 2.5 + 5i√3/2.

If z = 5(cos(θ) + i sin(θ)), what is the value of z when θ = π/3? (2023)

If z = 5(cos(θ) + i sin(θ)), what is the value of z when θ = π/3? (2023)

Step 1: Identify the given formula for z, which is z = 5(cos(θ) + i sin(θ)).
Step 2: Substitute θ with π/3 in the formula: z = 5(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 = 5(1/2 + i√3/2).
Step 6: Distribute the 5: z = 5 * (1/2) + 5 * (i√3/2).
Step 7: Calculate 5 * (1/2) = 2.5.
Step 8: Calculate 5 * (i√3/2) = (5i√3)/2.
Step 9: Combine the results: z = 2.5 + (5i√3)/2.

Complex Numbers – Understanding the representation of complex numbers in polar form and converting them to rectangular form.
Trigonometric Functions – Applying trigonometric values for specific angles, particularly π/3 in this case.