Carla Murphy

## Answered question

2022-01-02

The question is to prove:
$-\sqrt{{a}^{2}+{b}^{2}+2ab\mathrm{cos}\left(\alpha -\beta \right)}
I tried opening the brackets but Im

### Answer & Explanation

Foreckije

Beginner2022-01-03Added 32 answers

Note that
$S=Re\left(a{e}^{\alpha +\theta }+b{e}^{\beta +\theta }\right)$
You can imagine these two complex numbers as vectors in the Argand plane.The resultant of these two vectors, can be found from the rule of addition of vectors and would have magnitude equal to $\sqrt{{a}^{2}+{b}^{2}+2ab\mathrm{cos}\left(\alpha -\beta \right)}$. Now, the real part of this resultant vector would be S. This would be its cosine component, which lies between −1 and 1. Hence the result holds up.

lalilulelo2k3eq

Beginner2022-01-04Added 38 answers

This is equivalent to proving
${\left(a\mathrm{cos}\left(\alpha +\theta \right)+b\mathrm{cos}\left(\beta +\theta \right)\right)}^{2}\le {a}^{2}+{b}^{2}+2ab\mathrm{cos}\left(\alpha -\beta \right)$
$⇔{a}^{2}\left(1-{\mathrm{cos}}^{2}\left(\alpha +\theta \right)\right)+{b}^{2}\left(1-{\mathrm{cos}}^{2}\left(\beta +\theta \right)\right)+2ab\left[\mathrm{cos}\left(\alpha -\beta \right)-\mathrm{cos}\left(\theta +\alpha \right)\mathrm{cos}\left(\theta +\beta \right)\right]\ge 0$

$⇔{a}^{2}{\mathrm{sin}}^{2}\left(\alpha +\theta \right)+{b}^{2}{\mathrm{sin}}^{2}\left(\beta +\theta \right)+2ab\mathrm{sin}\left(\alpha +\theta \right)\mathrm{sin}\left(\beta +\theta \right)\ge 0$
$⇔{\left(a\mathrm{sin}\left(\alpha +\theta \right)+b\mathrm{sin}\left(\beta +\theta \right)\right)}^{2}\ge 0$, which is true
So, the given inequality holds

Vasquez

Expert2022-01-08Added 669 answers

Imagine two rotating vectors of lengths a,b and initial directions $\alpha ,\beta$ (with constant difference $\alpha -\beta$). By the cosine rule in the triangle formed by these three vectors, the length of the sum is
$\sqrt{{a}^{2}+{b}^{2}+2ab\mathrm{cos}\left(\alpha -\beta \right)}$
When the vectors rotate, the projection of the sum on the horizontal axis oscillates between plus and minus this value.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?