# Expressing a trig function as a complex expodential (HELP!)

frankR
The problem states:

Express Cos( &Theta;1 + &Theta;2 + &Theta;3) in terms of Sin(&Theta;k) and Cos(&Theta;k), k = 1, 2, 3, using the relation e+/-i*&Theta; = Cos(&Theta;) +/- i*Sin(&Theta;). [Hint: Use the product property of the exponential e.g., e(&Theta;1 + &Theta;2) = ei*&Theta;1ei*&Theta;2.]

I'm really confused by in terms of Sin(&Theta;k) and Cos(&Theta;k), k = 1, 2, 3, how does this apply to the problem?

I'm really lost, someone please steer me in the right direction.

Thanks,

Frank

Edit: Not sure what's wrong with my &theta ?

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## Answers and Replies

You need to put: - (Using A's instead of thetas)

cos(A1+A2+A3) + isin(A1+A2+A3)

into exponential form:

e^-i(A1+A2+A3)

then using the property of exponentials, express the above exponent as:

(e^-iA1)(e^-iA2)(e^-iA3)

Then put each exponential back into trig form and multiply everything out. Seperate the real and imaginary terms and equate cos(A1+A2+A3) to the real parts. This will contain both cos and sin terms.

Claude.

Originally posted by frankR
The problem states:

Express Cos( &theta;1 + &theta;2 + &theta;3) in terms of Sin(&theta;k) and Cos(&theta;k), k = 1, 2, 3, using the relation e+/-i*&theta; = Cos(&theta;) +/- Sinh(&theta;). [Hint: Use the product property of the exponential e.g., e(&theta;1 + &theta;2) = ei*&theta;1ei*&theta;2.]

I'm really confused by in terms of Sin(&theta;k) and Cos(&theta;k), k = 1, 2, 3, how does this apply to the problem?

I'm really lost, someone please steer me in the right direction.

Thanks,

Frank

Edit: Not sure what's wrong with my &theta ?
You forgot the ;

frankR
I actually trired that method, but was unsure if it was correct since the terms were so messy. I'll continue with that method and post my solution.

Thanks.

frankR
Is this how it's expressed:

&plusmn;Cos{&Theta;k} = e &plusmn; (&Theta;k) &plusmn; Sin{&Theta;k}, where k=1,2,3.

Thanks.

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±Cos{?k} = e ± (?k) ± Sin{?k}

No.

Cos(?) = (e^(i?)+e^(-i?))/2

Originally posted by frankR
I actually trired that method, but was unsure if it was correct since the terms were so messy. I'll continue with that method and post my solution.

Thanks.

Recall the formula for the double angle formula for cos:

cos(A+B) = cos(A)cos(B)-sin(A)sin(B)

Not the neatest answer, and that is just for the two angle case. Based on this alone, you should probably expect your answer to be a bit messy.

Claude.