Question: If z = 2(cos(π/3) + i sin(π/3)), what is z in rectangular form? (2022)
Options:
1 + √3i
2 + √3i
1 + 2i
2 + 2i
Correct Answer: 1 + √3i
Exam Year: 2022
Solution:
Using Euler\'s formula, z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + √3i.
If z = 2(cos(π/3) + i sin(π/3)), what is z in rectangular form? (2022)
Practice Questions
Q1
If z = 2(cos(π/3) + i sin(π/3)), what is z in rectangular form? (2022)
1 + √3i
2 + √3i
1 + 2i
2 + 2i
Questions & Step-by-Step Solutions
If z = 2(cos(π/3) + i sin(π/3)), what is z in rectangular form? (2022)
Step 1: Identify the given expression for z, which is z = 2(cos(π/3) + i sin(π/3)).
Step 2: Calculate cos(π/3). The value of cos(π/3) is 1/2.
Step 3: Calculate sin(π/3). The value of sin(π/3) is √3/2.
Step 4: Substitute the values of cos(π/3) and sin(π/3) into the expression for z: z = 2(1/2 + i√3/2).
Step 5: Distribute the 2 into the expression: z = 2 * (1/2) + 2 * (i√3/2).
Step 6: Simplify the expression: z = 1 + i√3.
Polar to Rectangular Conversion – The question tests the ability to convert a complex number from polar form (using trigonometric functions) to rectangular form (a + bi).
Euler's Formula – The use of Euler's formula, which relates complex exponentials to trigonometric functions, is essential for solving the problem.
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