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What is exp(ja) equal to, where j is the square root of unity?
- a)cos ja + jsin a
- b)sin a + jcos a
- c)cos j + a sin j
- d)cos a + jsin a
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
What is exp(ja) equal to, where j is the square root of unity?a)cos ja...
This is the corollary of DeMoivre/Euler’s Theorem.
Most Upvoted Answer
What is exp(ja) equal to, where j is the square root of unity?a)cos ja...
To understand the given expression, let's break it down step by step:
1. Square root of unity:
The square root of unity is denoted by 'j' and is defined as a complex number that satisfies the equation j^2 = -1.
2. Euler's formula:
Euler's formula is a mathematical expression that relates complex numbers to trigonometric functions. It states that for any real number 'x', the expression e^(jx) can be written as cos(x) + j*sin(x), where e is the base of the natural logarithm.
3. Applying Euler's formula to the given expression:
We are given exp(ja), where 'a' is a real number. Using Euler's formula, we can rewrite it as:
exp(ja) = cos(a) + j*sin(a)
4. Comparing the given options:
a) cos ja + jsin a
b) sin a + jcos a
c) cos j + a sin j
d) cos a + jsin a
Comparing the rewritten expression from step 3 with the given options, we can see that option 'd' matches the expression exactly:
exp(ja) = cos(a) + j*sin(a) = cos a + jsin a
Therefore, the correct answer is option 'd': cos a + jsin a.
In summary, when we have the expression exp(ja), where 'a' is a real number and 'j' is the square root of unity, it can be simplified to cos a + jsin a using Euler's formula.
1. Square root of unity:
The square root of unity is denoted by 'j' and is defined as a complex number that satisfies the equation j^2 = -1.
2. Euler's formula:
Euler's formula is a mathematical expression that relates complex numbers to trigonometric functions. It states that for any real number 'x', the expression e^(jx) can be written as cos(x) + j*sin(x), where e is the base of the natural logarithm.
3. Applying Euler's formula to the given expression:
We are given exp(ja), where 'a' is a real number. Using Euler's formula, we can rewrite it as:
exp(ja) = cos(a) + j*sin(a)
4. Comparing the given options:
a) cos ja + jsin a
b) sin a + jcos a
c) cos j + a sin j
d) cos a + jsin a
Comparing the rewritten expression from step 3 with the given options, we can see that option 'd' matches the expression exactly:
exp(ja) = cos(a) + j*sin(a) = cos a + jsin a
Therefore, the correct answer is option 'd': cos a + jsin a.
In summary, when we have the expression exp(ja), where 'a' is a real number and 'j' is the square root of unity, it can be simplified to cos a + jsin a using Euler's formula.
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