We can use this equation to solve this problem:
<span>t=p+<span>(p⋅r)</span></span>
Where:
t is the total price paid: 58.31 for this problem
p is the price before taxes: What we are solving for in this problem.
r is the tax or VAT rate: 19% for this problem. "Percent" or "%" means "out of 100" or "per 100", Therefore 19% can be written as <span>19100</span> or 0.19.
Substituting and solving for p gives:
<span>58.31=p+<span>(p⋅0.19)</span></span>
<span>58.31=1p+0.19p</span>
<span>58.31=<span>(1+0.19)</span>p</span>
<span>58.31=1.19p</span>
<span><span>58.311.19</span>=<span><span>1.19p</span>1.19</span></span>
<span>49=<span><span>1.19p</span>1.19</span></span>
<span>49=p</span>
<span>p=49</span>
Now, we can find how much the VAT was by taking 19% of 49:
<span>19%×49=0.19×49=9.31</span>
Another was to solve this is to subtract the price without the tax or VAT from the total price giving:
<span>58.31−49=<span>9.31</span></span>
![\bf sec(s)=cos(s)\qquad but\implies sec(\theta)=\cfrac{1}{cos(\theta)} \\\\\\ thus\cfrac{1}{cos(s)}=cos(s)\implies 1=cos^2(s)\implies \pm \sqrt{1}=cos(s) \\\\\\ \pm 1=cos(s)\impliedby \textit{now taking }cos^{-1}\textit{ to both sides} \\\\\\ cos^{-1}(\pm 1)=cos^{-1}[cos(s)]\implies cos^{-1}(\pm 1)=\measuredangle s](https://tex.z-dn.net/?f=%5Cbf%20sec%28s%29%3Dcos%28s%29%5Cqquad%20but%5Cimplies%20sec%28%5Ctheta%29%3D%5Ccfrac%7B1%7D%7Bcos%28%5Ctheta%29%7D%0A%5C%5C%5C%5C%5C%5C%0Athus%5Ccfrac%7B1%7D%7Bcos%28s%29%7D%3Dcos%28s%29%5Cimplies%201%3Dcos%5E2%28s%29%5Cimplies%20%5Cpm%20%5Csqrt%7B1%7D%3Dcos%28s%29%0A%5C%5C%5C%5C%5C%5C%0A%5Cpm%201%3Dcos%28s%29%5Cimpliedby%20%5Ctextit%7Bnow%20taking%20%7Dcos%5E%7B-1%7D%5Ctextit%7B%20to%20both%20sides%7D%0A%5C%5C%5C%5C%5C%5C%0Acos%5E%7B-1%7D%28%5Cpm%201%29%3Dcos%5E%7B-1%7D%5Bcos%28s%29%5D%5Cimplies%20cos%5E%7B-1%7D%28%5Cpm%201%29%3D%5Cmeasuredangle%20s)