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If z = 2e^(iπ/3), what is the value of z?
If z = 2e^(iπ/3), what is the value of z?
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Practice Questions
1 question
Q1
If z = 2e^(iπ/3), what is the value of z?
1 + i√3
2 + 0i
0 + 2i
2 - 2i
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z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + i√3.
Questions & Step-by-step Solutions
1 item
Q
Q: If z = 2e^(iπ/3), what is the value of z?
Solution:
z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + i√3.
Steps: 8
Show Steps
Step 1: Identify the given value of z, which is z = 2e^(iπ/3).
Step 2: Use Euler's formula, which states that e^(ix) = cos(x) + i sin(x).
Step 3: Substitute π/3 into Euler's formula: e^(iπ/3) = cos(π/3) + i sin(π/3).
Step 4: Calculate cos(π/3) and sin(π/3). The values are cos(π/3) = 1/2 and sin(π/3) = √3/2.
Step 5: Substitute these values back into the equation: e^(iπ/3) = 1/2 + i(√3/2).
Step 6: Multiply the entire expression by 2: z = 2(1/2 + i(√3/2)).
Step 7: Distribute the 2: z = 2 * 1/2 + 2 * i(√3/2).
Step 8: Simplify the expression: z = 1 + i√3.
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