From \(\sin{\alpha}=\dfrac{4}{5}\),
$$ \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5} $$
From \(\tan{\alpha}=\dfrac{4}{3}\),
$$ \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3} $$
$$ \cos{\alpha} = \frac{\text{adjacent}}{\text{hypotenuse}} = \boxed{\frac{3}{5}} $$
$$ \tan{\alpha} = \frac{\sin{\alpha}}{\cos{\alpha}} $$
$$ \cos{\alpha} = \frac{\sin{\alpha}}{\tan{\alpha}} = \frac{\frac{4}{5}}{\frac{4}{3}}$$
$$ \cos{\alpha} = \frac{4}{5}\cdot \frac{3}{4} = \boxed{\frac{3}{5}} $$
You can find the value of \(\alpha\) using your calculator.
$$ \sin{\alpha} = \frac{4}{5} $$
$$ \alpha = \arcsin{\frac{4}{5}} $$
$$ \cos{\alpha} = \cos{\left(\arcsin{\frac{4}{5}}\right)} = 0.6 = \boxed{\frac{3}{5}} $$