Solve for θₐ s_a-s_b=r_atheta_a-r_btheta_b

$s_{a} - s_{b} = r_{a} θ_{a} - r_{b} θ_{b}$

Step 1

Rewrite the equation as $r_{a} θ_{a} - r_{b} θ_{b} = s_{a} - s_{b}$ .

$r_{a} θ_{a} - r_{b} θ_{b} = s_{a} - s_{b}$

Step 2

Add $r_{b} θ_{b}$ to both sides of the equation.

$r_{a} θ_{a} = s_{a} - s_{b} + r_{b} θ_{b}$

Step 3

Divide each term in $r_{a} θ_{a} = s_{a} - s_{b} + r_{b} θ_{b}$ by $r_{a}$ and simplify.

Divide each term in $r_{a} θ_{a} = s_{a} - s_{b} + r_{b} θ_{b}$ by $r_{a}$ .

$\frac{r_{a} θ_{a}}{r_{a}} = \frac{s_{a}}{r_{a}} + \frac{- s_{b}}{r_{a}} + \frac{r_{b} θ_{b}}{r_{a}}$

Simplify the left side.

Cancel the common factor of $r_{a}$ .

Cancel the common factor.

$\frac{r_{a} θ_{a}}{r_{a}} = \frac{s_{a}}{r_{a}} + \frac{- s_{b}}{r_{a}} + \frac{r_{b} θ_{b}}{r_{a}}$

Divide $θ_{a}$ by $1$ .

$θ_{a} = \frac{s_{a}}{r_{a}} + \frac{- s_{b}}{r_{a}} + \frac{r_{b} θ_{b}}{r_{a}}$

Simplify the right side.

Move the negative in front of the fraction.

$θ_{a} = \frac{s_{a}}{r_{a}} - \frac{s_{b}}{r_{a}} + \frac{r_{b} θ_{b}}{r_{a}}$