1. **State the problem:** We have a right triangle with a vertical leg of length 7, a hypotenuse labeled $w$, and an angle of $74^\circ$ opposite the vertical leg. We want to find expressions that represent the length of $w$. 2. **Recall the trigonometric definitions:** In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse, and the sine of an angle is the ratio of the opposite side to the hypotenuse. 3. **Identify the sides relative to the $74^\circ$ angle:** - Opposite side to $74^\circ$ is the vertical leg of length 7. - Hypotenuse is $w$. 4. **Write the sine relation:** $$\sin(74^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7}{w}$$ 5. **Solve for $w$ using sine:** $$w = \frac{7}{\sin(74^\circ)}$$ 6. **Write the cosine relation:** The adjacent side to $74^\circ$ is the base (not given), so cosine is not directly useful here for $w$ in terms of 7. 7. **Check other expressions:** - $7 \cdot \cos(16^\circ)$: $16^\circ$ is the complement of $74^\circ$, but this expression does not relate $w$ to 7 correctly. - $7 / \cos(74^\circ)$: Using cosine for the adjacent side, but 7 is opposite side, so this is incorrect. - $7 / \sin(16^\circ)$: $16^\circ$ is complementary angle, and $\sin(16^\circ) = \cos(74^\circ)$, so this is equivalent to $7 / \cos(74^\circ)$, which is incorrect. - $7 / \cos(16^\circ)$: This is $7 / \sin(74^\circ)$, which matches the correct expression for $w$. - $7 \cdot \sin(74^\circ)$: This equals the opposite side times sine, which is not $w$. 8. **Conclusion:** The two correct expressions for $w$ are: $$w = \frac{7}{\sin(74^\circ)}$$ and equivalently $$w = \frac{7}{\cos(16^\circ)}$$ **Final answer:** $w = \frac{7}{\sin(74^\circ)}$ and $w = \frac{7}{\cos(16^\circ)}$