Question

1. noah missed class, and clare is helping him use the table to approximate the angle measures that have the ratios listed. clare says, \you can use the right triangle table to figure this out.\ noah notices that some of the ratios are the same in each row. estimate the angles and explain why some of the values are repeated.

angle	adjacent leg ÷ hypotenuse	opposite leg ÷ hypotenuse	opposite leg ÷ adjacent leg
	0.174	0.985	5.671

explain why there are equivalent values but in different columns for both angles?

Explanation:

These two angles are complementary (they add up to 90°), meaning one is the acute angle $\theta$ and the other is $90^\circ - \theta$ in a right triangle. For angle $\theta$, the adjacent leg is the opposite leg for $90^\circ - \theta$, and vice versa. The cosine of $\theta$ (adjacent/hypotenuse) equals the sine of $90^\circ - \theta$ (opposite/hypotenuse for the complementary angle), and the sine of $\theta$ equals the cosine of $90^\circ - \theta$. The tangent of $90^\circ - \theta$ is the reciprocal of the tangent of $\theta$, which matches the 0.176 and 5.671 values (since $\frac{1}{0.176} \approx 5.671$).

Answer:

The two angles are complementary (sum to $90^\circ$). In a right triangle, the adjacent side of one acute angle is the opposite side of the other acute angle, so their sine and cosine ratios swap, and their tangent ratios are reciprocals. This causes the equivalent values to appear in different columns for the two angles.