1. **State the problem:** We have a right triangle with angle $A = 50^\circ$, the side adjacent to angle $A$ is 16, and we need to find the side opposite angle $A$ (denoted $a$) and the hypotenuse $c$. 2. **Recall the trigonometric definitions:** - The side adjacent to angle $A$ is related to the hypotenuse by $\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}$. - The side opposite angle $A$ is related to the hypotenuse by $\sin A = \frac{\text{opposite}}{\text{hypotenuse}}$. - The side opposite angle $A$ is related to the adjacent side by $\tan A = \frac{\text{opposite}}{\text{adjacent}}$. 3. **Find the hypotenuse $c$ using cosine:** $$\cos 50^\circ = \frac{16}{c}$$ Multiply both sides by $c$: $$c \cos 50^\circ = 16$$ Divide both sides by $\cos 50^\circ$: $$c = \frac{16}{\cos 50^\circ}$$ 4. **Calculate $c$ numerically:** $$c = \frac{16}{\cos 50^\circ} = \frac{16}{0.6428} \approx 24.9$$ 5. **Find the side opposite angle $A$, $a$, using tangent:** $$\tan 50^\circ = \frac{a}{16}$$ Multiply both sides by 16: $$16 \tan 50^\circ = a$$ 6. **Calculate $a$ numerically:** $$a = 16 \times \tan 50^\circ = 16 \times 1.1918 \approx 19.1$$ 7. **Summary of answers:** - Angle $A = 50^\circ$ (given) - Side $a \approx 19.1$ - Hypotenuse $c \approx 24.9$