Use the image below to answer the following question. What relationship do the ratios of sin x° and cos y° share?

Question

A right triangle is shown. The two angles that are not 90 degrees are marked x and y. The leg across from angle y measuring 12, another leg across from angle x measuring 5, and the hypotenuse measuring 13.

The ratios are opposites (negative five thirteenths and five thirteenths).
The ratios are both identical (five thirteenths and five thirteenths).
The ratios are reciprocals (five thirteenths and thirteen fifths).
The ratios are both negative (negative thirteen fifths and negative thirteen fifths).

Answer 1

To determine the relationship between the ratios of sin x° and cos y°, we first need to understand the definitions of these trigonometric functions in the context of the given right triangle.

1. The sine of angle x (sin x°) is the ratio of the length of the opposite side to the hypotenuse. The leg opposite angle x is 5, and the hypotenuse is 13. Therefore: \[ \sin x° = \frac{5}{13} \]

2. The cosine of angle y (cos y°) is the ratio of the length of the adjacent side to the hypotenuse. The leg adjacent to angle y is 12, and the hypotenuse is also 13. Therefore: \[ \cos y° = \frac{12}{13} \]

Now, to analyze the relationship between sin x° and cos y°, we note:

• The two angles are complementary in a right triangle, meaning \( x + y = 90° \). Therefore, the sine of one angle is equal to the cosine of the other: \[ \sin x° = \cos(90° - x) = \cos y° \]

Since we have \( \sin x° = \frac{5}{13} \) and \( \cos y° = \frac{12}{13} \), these values are not the same but represent the sine and cosine of complementary angles.

Since there is no option that states that they are equal directly, nor does their relationship imply that they are reciprocals or negatives of one another in terms of their values, the best interpretation is that:

They represent related ratios of complementary angles but are not directly relational in the options given. If forced to choose, the proportions and their trigonometric relationships suggest they are not identical, negative, or reciprocals. Thus, one might note their derivation from similar triangle relationships. However, none of the provided options correctly defines the relationship between sin x° and cos y°.

Therefore, the closest approach to express might be:

• The ratios are not directly equivalent or negated. Still, the sine of one angle describes the cosine of the complementary angle, leading to the deduction of ratios representing different triangle sides.

But purely following the options, if limited, highlight as above but state they are simply ratios of sine and cosine without direct negation or reciprocation attributes.