Question: If z = 2(cos(π/4) + i sin(π/4)), find the rectangular form of z.
Options:
√2 + √2i
2 + 2i
1 + i
0 + 0i
Correct Answer: √2 + √2i
Solution:
z = 2(cos(π/4) + i sin(π/4)) = 2(√2/2 + i√2/2) = √2 + √2i.
If z = 2(cos(π/4) + i sin(π/4)), find the rectangular form of z.
Practice Questions
Q1
If z = 2(cos(π/4) + i sin(π/4)), find the rectangular form of z.
√2 + √2i
2 + 2i
1 + i
0 + 0i
Questions & Step-by-Step Solutions
If z = 2(cos(π/4) + i sin(π/4)), find the rectangular form of z.
Step 1: Start with the given equation z = 2(cos(π/4) + i sin(π/4)).
Step 2: Calculate cos(π/4) and sin(π/4). Both are equal to √2/2.
Step 3: Substitute the values of cos(π/4) and sin(π/4) into the equation: z = 2(√2/2 + i√2/2).
Step 4: Distribute the 2: z = 2 * (√2/2) + 2 * (i√2/2).
Step 5: Simplify the equation: z = √2 + √2i.
Step 6: The rectangular form of z is √2 + √2i.
Polar to Rectangular Conversion – The question tests the ability to convert a complex number from polar form (using cosine and sine) to rectangular form (a + bi).
Trigonometric Values – It requires knowledge of the trigonometric values of common angles, specifically π/4.
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