# Rearranging redshift formula for v

by Pretty Pony
Tags: formula, rearranging, redshift
 P: 8 Hey there I would just like some help in rearranging the formula: $$z=\frac{1+\frac{v\cos \theta}{c}}{\sqrt{1-\frac{v^2}{c^2}}}-1 [\tex] for v. Thanks, PP  Math Emeritus Sci Advisor Thanks PF Gold P: 38,904 That is [tex]z= \frac{1+ \frac{v cos(\theta)}{c}}{\sqrt{1- \frac{v^2}{c^2}}}- 1$$ You solve an equation by "undoing" what was done to the unknown, step by step. The last thing done there is "-1" so we undo that by adding 1 to both sides: $$z+ 1= \frac{1+ \frac{v cos(\theta)}{c}}{\sqrt{1- \frac{v^2}{c^2}}}$$ Now, multiply on both sides by the denominator: $$(z+1)\left(\sqrt{1- \frac{v^2}{c^2}}\right)= 1+ \frac{v cos(\theta)}{c}$$ Get rid of that square root by squaring both sides (which might introduce false solution- you will need to check for that after finding v): $$(z+1)^2\left(1- \frac{v^2}{c^2}\right)= 1+ 2\frac{v cos(\theta)}{c}+ \frac{v^2 cos^2(\theta)}{c^2}$$ Now that is a quadratic equation in v. Combining like terms, $$\left(\frac{1}{cos^2(\theta)}+ \frac{(z+1)^2}{c^2}\right)v^2+ \frac{2vcos(\theta)}{c}v+ 1- (z+1)^2= 0$$ which can be solved using the quadratic formula.
 P: 8 Thanks for your help! :)

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