Proving the Trigonometric Identity Sin(a i*b) x iy

Understanding the relationship between trigonometric and hyperbolic functions can provide deep insights into the behavior of complex numbers. One interesting identity involves the expression sin(a ib), where a and b are real numbers, and ix and iy are the real components.

Step-by-Step Solution

The given expression is:

sin(a ib) x iy

First, let's reformat the original problem for clarity:

Given sinaib x iy, it can be expanded as:

frac{x^2}{sin^2a} - frac{y^2}{cos^2a} 1

And,

frac{x^2}{cosh^2b} cdot frac{y^2}{sinh^2b} 1

Step 1: Expand the Addition Formula

Let's start by expanding the addition formula for sine:

sin(a ib) sin a cos ib sin ib cos a

Next, we need to evaluate cos ib and sin ib using their definitions:

cos ib frac{e^{iib} e^{-iib}}{2} frac{e^{-b}e^b e^be^{-b}}{2} cosh b

sin ib frac{e^{iib} - e^{-iib}}{2i} i frac{e^b - e^{-b}}{2} isinh b

Substituting these into our expression gives:

sin(a ib) sin a cosh b icos a sinh b

Thus, we can identify:

x sin a cosh b

y cos a sinh b

Verification of the Hyperbolic Identity

To verify that these expressions for x and y satisfy the given equations:

frac{x^2}{sin^2a} - frac{y^2}{cos^2a} 1

And,

frac{x^2}{cosh^2b} cdot frac{y^2}{sinh^2b} 1

Verification 1:

Substitute x and y into the first equation:

frac{(sin a cosh b)^2}{sin^2a} - frac{(cos a sinh b)^2}{cos^2a}

Simplifying the numerator and denominator:

[ frac{sin^2a cosh^2b}{sin^2a} - frac{cos^2a sinh^2b}{cos^2a} ]

[ cosh^2b - sinh^2b ]

Using the hyperbolic identity cosh^2b - sinh^2b 1, we get:

[ 1 ]

The first equation is satisfied.

Verification 2:

Substitute x and y into the second equation:

frac{(sin a cosh b)^2}{cosh^2b} cdot frac{(cos a sinh b)^2}{sinh^2b}

Simplifying the numerator and denominator:

[ frac{sin^2acosh^2b}{cosh^2b} cdot frac{cos^2asinh^2b}{sinh^2b} ]

[ sin^2a cdot cos^2a ]

Since sin^2a cdot cos^2a 1 (this follows from the Pythagorean identity for sine and cosine), we get:

[ 1 ]

The second equation is also satisfied.

Conclusion

The identity sin(a ib) sin a cosh b icos a sinh b has been proven by equating the components and verifying them against the given hyperbolic and trigonometric identities. This elegant result demonstrates the interconnectedness of trigonometric and hyperbolic functions, offering both theoretical and practical applications in mathematics and physics.

Related Keywords: Trigonometric identity, complex numbers, hyperbolic functions