For an angle with measure $\alpha$ in a right triangle, $\sin{\alpha}=\dfrac{4}{5}$ and $\tan{\alpha}=\dfrac{4}{3}$. What is the value of $\cos{\alpha}$ ?

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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.

$$\sin{\alpha} = \frac{4}{5}$$ $$\alpha = \arcsin{\frac{4}{5}}$$ $$\cos{\alpha} = \cos{\left(\arcsin{\frac{4}{5}}\right)} = 0.6 = \boxed{\frac{3}{5}}$$