Answer
$C \gt B \gt A$. Therefore the largest angle is $\angle C = 68.7˚$.
Work Step by Step
1. Use the cosine formula to find $\angle C$
$c^{2} = a^{2} + b^{2} - 2abcos(C)$
$(26)^{2} = (22)^{2} + (24)^{2} - 2(22)(24)cos(C)$
$676 = 1060 - 1056cos(C)$
$-384 = -1056cos(C)$
$0.363636... = cos(C)$
by GDC / calculator
$C = 68.6763...˚$
$C = 68.7˚$
2. Use the sine formula to find $\angle A$ and $\angle B$
$\frac{sin(A)}{22}= \frac{sin(68.6763...)}{26}$
$sin(A) = \frac{22sin(68.67...)}{26}$
by GDC / calculator
$sin(A) = 0.78822...$
$A = 52.0201...˚$
$A = 52.0˚$
$\frac{sin(B)}{24} = \frac{sin(68.6763...)}{26}$
$sin(B) = \frac{24sin(68.6763...)}{26}$
$sin(B) = 0.85988....$
by GDC / calculator
$B = 59.3035...˚$
$B = 59.3˚$
$C \gt B \gt A$. Therefore the largest angle is $\angle C = 68.7˚$.
