# Exponential forms of cos and sin

• Dough
In summary, the conversation discusses the use of exponential forms and trigonometric identities in proving equations in Modern Engineering Mathematics by Glyn James. The conversation also touches on the use of imaginary numbers in engineering and the confusion between i and j.
Dough
hi, my question is from Modern Engineering Mathematics by Glyn James

pg 177 # 17a

Using the exponential forms of cos(theta) and sin(theta) given in (3.11a, b), prove the following trigonometric identities:
a) sin(x + y) = sin(x)cos(y) + cos(x)sin(y)

and 3.11a is:
cos(x) = 0.5*[ e^(jx) + e^(-jx) ] where x= theta
and 3.11b is:
sin(x) = 0.5j*[ e^(jx) - e^(-jx) ] where x= theta

i've gotten to the point where i have
[ e^j(x+y) + e^j(x-y) -e^j(y-x) -e^-j(x + y) ] / 2j

3.11b should be

sin(x) = 0.5*[ e^(jx) - e^(-jx) ]/j

or

sin(x) = -0.5j*[ e^(jx) - e^(-jx) ]

What you want to do here is start with the expression 3.11b of sin(x+y), then use the property of the exponential that exp(x+y)=exp(x)exp(y) and then transform the exponentials back into sin and cos form. Then the real part of that is sin(x+y) (and the imaginary part equals 0).

whoops i meant for the j to be on the bottom for 3.11b

its like
[1/(2j)][ e^(jx) - e^(-jx) ]

weee i got it thanks for the help!

Oh, those engineers and their jmagjnary numbers!

hello my physics chums,
hows your equations looking? I want somebody to love me.

kevmyster

HallsofIvy said:
Oh, those engineers and their jmagjnary numbers!

Yes that's a pain in the ..., always confusion between i and j.

HallsofIvy said:
Oh, those engineers and their jmagjnary numbers!

In electrical engineering, i means electric current so imaginary numbers are written as j. I remember j was first introduced in the "Elementary linear circuit analysis" class and then all engineering classes use j instead of i. Yes, when reading mathematics or physics paper, I need to switch to i mode. And, mathematicians and physicists write Fourier transform in a confusing form

## 1. What is the difference between exponential forms of cos and sin?

The exponential form of cos and sin refers to representing these trigonometric functions using complex numbers. The main difference is that cos involves an exponential function with imaginary numbers, while sin involves an exponential function with a real number.

## 2. How are exponential forms of cos and sin useful in mathematics?

Exponential forms of cos and sin are useful in many areas of mathematics, particularly in complex analysis and calculus. They allow for the simplification of complex trigonometric expressions and the use of techniques such as Euler's formula to solve problems involving these functions.

## 3. Can exponential forms of cos and sin be used to solve real-world problems?

Yes, exponential forms of cos and sin can be used to solve real-world problems, such as in electrical engineering and physics. These functions have many practical applications, and the use of their exponential forms can make calculations more efficient.

## 4. Are there any limitations to using exponential forms of cos and sin?

One limitation of using exponential forms of cos and sin is that they are not always intuitive to work with, especially for those who are not familiar with complex numbers. Additionally, they may not always be the most efficient method for solving certain problems, and other techniques may be more suitable.

## 5. Can exponential forms of cos and sin be converted back to their trigonometric forms?

Yes, exponential forms of cos and sin can be converted back to their trigonometric forms using Euler's formula. This allows for the simplification of complex expressions and the ability to solve problems in their original form.

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