In one line
Expand with the addition formula and then use \(\cos iy=\cosh y\), \(\sin iy=i\sinh y\); for example \(\sin(x+iy)=\sin x\cosh y+i\cos x\sinh y\).
Explained simply
A complex function value is like a two-part answer: a real part and an imaginary part. Separation means writing it as \(u+iv\) with \(u\) and \(v\) real. You already know \(\sin(A+B)=\sin A\cos B+\cos A\sin B\). Put \(A=x\), \(B=iy\), and the bridge \(\sin iy=i\sinh y\) does the rest.
Technically, for real \(x,y\): \(\sin(x+iy)=\sin x\cos iy+\cos x\sin iy=\sin x\cosh y+i\cos x\sinh y\). The real part is \(\sin x\cosh y\) and the imaginary part is \(\cos x\sinh y\).
How it works
- Write the addition formula for the function.
- Replace \(\cos iy=\cosh y\), \(\sin iy=i\sinh y\) (or \(\cosh iy=\cos y\), \(\sinh iy=i\sin y\)).
- Collect terms without \(i\) (real part) and with \(i\) (imaginary part).
- In "If \(\sin(A+iB)=x+iy\), prove..." problems, equate real and imaginary parts to get two real equations, then eliminate using \(\sin^2+\cos^2=1\) or \(\cosh^2-\sinh^2=1\).
Key formula
- \(\sin(x+iy)=\sin x\cosh y+i\cos x\sinh y\)
- \(\cos(x+iy)=\cos x\cosh y-i\sin x\sinh y\)
- \(\sinh(x+iy)=\sinh x\cos y+i\cosh x\sin y\)
- \(\cosh(x+iy)=\cosh x\cos y+i\sinh x\sin y\)
\(x,y\) are real numbers.
Quick example
If \(\sin(\alpha+i\beta)=x+iy\), prove \(\dfrac{x^2}{\cosh^2\beta}+\dfrac{y^2}{\sinh^2\beta}=1\).
Expand: \(\sin\alpha\cosh\beta+i\cos\alpha\sinh\beta=x+iy\). Equate: \(x=\sin\alpha\cosh\beta\), \(y=\cos\alpha\sinh\beta\). Then \(\dfrac{x^2}{\cosh^2\beta}+\dfrac{y^2}{\sinh^2\beta}=\sin^2\alpha+\cos^2\alpha=1\).
Eliminating \(\alpha\) instead, \(\dfrac{x^2}{\sin^2\alpha}-\dfrac{y^2}{\cos^2\alpha}=\cosh^2\beta-\sinh^2\beta=1\). These are an ellipse and a hyperbola.
Common mistakes
- Sign of the imaginary part of \(\cos(x+iy)\): it is minus, because \(\cos(A+B)\) has a minus.
- Equating real and imaginary parts before the expression is fully in \(u+iv\) form.
- Writing \(\cos iy=\cos y\) instead of \(\cosh y\).
How MU asks it
Very frequent 5/6-mark question type: "If \(\cos(\alpha+i\beta)=x+iy\), prove \((1+x)^2+y^2=(\cosh\beta+\cos\alpha)^2\)" or "If \(\sin(\theta+i\phi)=\cos\alpha+i\sin\alpha\), prove \(\cos^2\theta=\pm\sin\alpha\)". For full marks show the expansion, the use of the circular-hyperbolic relations, the two equated equations, and the elimination step clearly.