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Chapter 5: Problem 111

(a) Write a proof of the formula for \(\sin (u+v)\). (b) Write a proof of the formula for \(\sin (u-v)\).

Short Answer

Expert verified
The formula for \(\sin (u+v)\) is \(\sin(u) \cos(v) + \cos(u) \sin(v)\) and the formula for \(\sin (u-v)\) is \(\sin(u) \cos(v) - \cos(u) \sin(v)\). These identities are derived from Euler鈥檚 formula.

Step by step solution

01

Review of Euler's Formula

Euler鈥檚 formula is \(e^{ix}= \cos(x) + i \sin(x)\). It ties together the exponential function with the cosine and sine functions when using complex numbers.
02

Apply Euler's formula for \(u+v\)

Substitute \(u+v\) into Euler鈥檚 formula. This yields the equation \(e^{i(u+v)} = \cos(u+v) + i \sin(u+v)\).
03

Simplify the exponential

Using the rule of exponents which states that \(e^{a+b} = e^a * e^b\), the above equation can be simplified to \(e^{iu} * e^{iv} = \cos(u+v) + i \sin(u+v)\).
04

Use Euler's formula again

Substitute back into Euler鈥檚 formula for each of the components, known as \(e^{iu}\) and \(e^{iv}\). This yields \((\cos(u) + i \sin(u)) * (\cos(v) + i \sin(v)) = \cos(u+v) + i \sin(u+v)\). After expanding the right side and matching real to real and imaginary to imaginary parts, this will give us \(\sin(u+v) = \sin(u) \cos(v) + \cos(u) \sin(v)\).
05

Apply Euler's formula for \(u-v\)

Substitute \(u-v\) into Euler鈥檚 formula. This yields the similar equation to Step 2: \(e^{i(u-v)} = \cos(u-v) + i \sin(u-v)\).
06

Simplify the exponential and substitute back

Like in Step 3 and 4, use the rule of exponents and substitute back into Euler's formula. This will results in the equation \(\sin(u-v) = \sin(u) \cos(v) - \cos(u) \sin(v)\).

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