Hello StackExchange people,
I have an issue. I need to create a formula to find degrees and size at a math issue.
So I have $0-359$ degrees and a length of $2$ lines, and I need to get the length and degrees of the third vector (the resulting) with a single formula. I believe (not sure though) $\sin, \cos$ and/or $\tan$ might be required. Am I right?
The reason I'd need just a formula is because this is for programming an applet I'm making and I'm having problems with this part. Having this would make me able to finish my program.
Represent you vectors by complex numbers. Then your two vectors are $a_1e^{i\theta_1}$ and $a_2e^{i\theta_2}$. But to add them you need to get them into Cartesian form. Use Euler's formula for that: $re^{i\theta} = r(\cos(\theta) + i\sin(\theta))$. Then add up the real parts and imaginary parts. And finally, if you need the angle and length back you can convert them back into polar form via $r= \sqrt{x^2+y^2}$ and $\theta = \operatorname{atan2}(y,x)$, where $x$ is the real part and $y$ is the imaginary part.
Let's take an example. Say your two vectors are $a=2e^{3i}$ (meaning a vector with length $2$ and an angle of $3\ \text{rad}$) and $b=4.2e^{2i}$ (meaning a vector with length $4.2$ and an angle of $2\ \text{rad}$). Then $a=2\cos(3) + 2i\sin(3)$ and $b=4.2\cos(2)+4.2i\sin(2)$.
So $$a+b = (2\cos(3)+4.2\cos(2)) + (2\sin(3)+4.2\sin(2))i$$. Converting that back to polar form we get $r=\sqrt{(2\cos(3)+4.2\cos(2))^2 + (2\sin(3)+4.2\sin(2))^2}\approx 5.54$ and $\theta = \operatorname{atan2}(2\sin(3)+4.2\sin(2), 2\cos(3)+4.2\cos(2)) \approx 2.31\ \text{rad}$. So your final vector is $$a+b = \require{enclose}\enclose{box}{5.54e^{2.31 i}}$$