Question

Find the distance between points A and B. A (a cos alpha , a sin alpha), B (a cos beta, a sin beta)​

Answer 1

Answer:

$\sf AB=2\left|a\sin\dfrac{\alpha-\beta}{2}\right|$

Step-by-step explanation:

Given :

Two points and asked to find distance between them :

We know :

$\sf D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

$\sf AB=\sqrt{(a \cos\beta -a\cos\alpha)^2+(a\sin\beta -a\sin\alpha)^2}$

$\sf AB=\sqrt{a^2( \cos\beta -\cos\alpha)^2+a^2(\sin\beta -\sin\alpha)^2}$

$\sf AB=a\sqrt{( \cos\beta -\cos\alpha)^2+(\sin\beta -\sin\alpha)^2}$

$\sf AB=a\sqrt{\cos^2\beta +\cos^2\alpha-2\cos\beta.\cos\alpha+\sin^2\beta +\sin^2\alpha-sin\beta.\sin\alpha}$

$\sf AB=a\sqrt{1+1-2\cos\beta.\cos\alpha-2sin\beta.\sin\alpha}$

$\sf AB=a\sqrt{2(1-\cos\{\alpha-\beta\}}$

$\sf AB=a\sqrt{4 \sin^2.\left(\dfrac{\alpha-\beta}{2}\right) }$

$\sf AB=2\left|a\sin\dfrac{\alpha-\beta}{2}\right|$

Therefore we get required answer.

Answer 2

$\huge\bold\green{Question}$

Find the distance between points A and B. A (a cos alpha , a sin alpha), B (a cos beta, a sin beta)

$\huge\bold\green{Answer}$

According to the question we have given :-

Two points and they asked to find distance between the two given points

So, as we know that -

$\begin{lgathered}\implies\tt D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\\\\end{lgathered}$

$\begin{lgathered}\implies\tt AB=\sqrt{(a \cos\beta -a\cos\alpha)^2+(a\sin\beta -a\sin\alpha)^2}\\\\\end{lgathered}$

$\begin{lgathered}\implies\tt AB=\sqrt{a^2( \cos\beta -\cos\alpha)^2+a^2(\sin\beta -\sin\alpha)^2}\\\\\end{lgathered}$

$\begin{lgathered}\implies\tt AB=a\sqrt{( \cos\beta -\cos\alpha)^2+(\sin\beta -\sin\alpha)^2}\\\\\end{lgathered}$

$\begin{lgathered}\implies\tt AB=a\sqrt{\cos^2\beta +\cos^2\alpha-2\cos\beta.\cos\alpha+\sin^2\beta +\sin^2\alpha-sin\beta.\sin\alpha}\\\\\end{lgathered}$

$\begin{lgathered}\implies\tt AB=a\sqrt{1+1-2\cos\beta.\cos\alpha-2sin\beta.\sin\alpha}\\\\\end{lgathered}$

On further solving it , we get

$\begin{lgathered}\implies\ttAB=a\sqrt{2(1-\cos\{\alpha-\beta\}}\\\\\end{lgathered}$

$\begin{lgathered}\implies\tt AB=a\sqrt{4 \sin^2.\left(\dfrac{\alpha-\beta}{2}\right) }\\\\\end{lgathered}$

So, by solving this we get the correct answer

$\begin{lgathered}\red\implies\tt AB=2\left|a\sin\dfrac{\alpha-\beta}{2}\right|\\\\\end{lgathered}$