  Let $k$ be a field and $T$ be an algebraic $k$-torus. In 1969, over a global
field $k$, Voskresenskii proved that there exists an exact sequence $0\to
A(T)\to H^1(k,{\rm Pic}\,\overline{X})^\vee\to Sha(T)\to 0$ where $A(T)$ is the
kernel of the weak approximation of $T$, $Sha(T)$ is the Shafarevich-Tate group
of $T$, $X$ is a smooth $k$-compactification of $T$,
$\overline{X}=X\times_k\overline{k}$, ${\rm Pic}\,\overline{X}$ is the Picard
group of $\overline{X}$ and $\vee$ stands for the Pontryagin dual. On the other
hand, in 1963, Ono proved that for the norm one torus $T=R^{(1)}_{K/k}(G_m)$ of
$K/k$, $Sha(T)=0$ if and only if the Hasse norm principle holds for $K/k$.
First, we determine $H^1(k,{\rm Pic}\, \overline{X})$ for algebraic $k$-tori
$T$ up to dimension $5$. Second, we determine $H^1(k,{\rm Pic}\, \overline{X})$
for norm one tori $T=R^{(1)}_{K/k}(G_m)$ with $[K:k]=n\leq 15$ and $n\neq 12$.
We also show that $H^1(k,{\rm Pic}\, \overline{X})=0$ for
$T=R^{(1)}_{K/k}(G_m)$ when the Galois group of the Galois closure of $K/k$ is
the Mathieu group $M_n\leq S_n$ with $n=11,12,22,23,24$. Third, we give a
necessary and sufficient condition for the Hasse norm principle for $K/k$ with
$[K:k]=n\leq 15$ and $n\neq 12$. As applications of the results, we get the
group $T(k)/R$ of $R$-equivalence classes over a local field $k$ via
Colliot-Th\'{e}l\`{e}ne and Sansuc's formula and the Tamagawa number $\tau(T)$
over a number field $k$ via Ono's formula
$\tau(T)=|H^1(k,\widehat{T})|/|Sha(T)|$.
