Simplify a e^(-4i) + b e^(4i) into trigonometric form using Euler's formula.

Step-by-step solution

1. 1

   State Euler's formula and note that $4$ is in radians.

   $e^{i\theta}=\cos\theta+i\sin\theta$

   Here $\theta=\pm4$ radians - a fixed number, not a variable, so $\cos 4$ and $\sin 4$ are just constants ($\cos4=-0.6536$, $\sin4=-0.7568$).

2. 2

   Expand each exponential. Using $\cos(-\theta)=\cos\theta$ and $\sin(-\theta)=-\sin\theta$:

   $e^{-4i}=\cos 4-i\sin 4,\qquad e^{4i}=\cos 4+i\sin 4$

   The two share the same cosine but opposite sines - that asymmetry is what produces the $b-a$ below.

3. 3

   Substitute and distribute the coefficients.

   $a\,e^{-4i}+b\,e^{4i}=a(\cos 4-i\sin 4)+b(\cos 4+i\sin 4)$

4. 4

   Group real and imaginary parts. Collecting $\cos 4$ terms and $\sin 4$ terms separately:

   $=(a+b)\cos 4+i\,(b-a)\sin 4$

   So the real part carries the sum $a+b$ and the imaginary part carries the difference $b-a$ - note the order, $b$ minus $a$.

5. 5

   Check the two special cases. If $b=a$ the imaginary part vanishes and the result is $2a\cos4$, matching $a(e^{4i}+e^{-4i})=2a\cos4$ ✓. If $b=-a$ the real part vanishes, giving $-2ai\sin4$, matching $a(e^{-4i}-e^{4i})=-2ai\sin4$ ✓.

6. 6

   Verify numerically. With $a=1.3$, $b=-0.7$: direct evaluation gives $-0.392186+1.513605i$, and $(a+b)\cos4+i(b-a)\sin4=0.6(-0.65364)+i(-2)(-0.75680)=-0.392186+1.513605i$ ✓.