To answer this question, we need to understand the relationship between trigonometric functions, especially cosine and sine, and the concepts of congruent, complementary, corresponding, and supplementary angles.
Firstly, congruent angles are angles that have the same measure. For example, if angle X is 30 degrees and angle Y is also 30 degrees, then X and Y are congruent.
The trigonometric functions cosine (cos) and sine (sin) of an angle in a right-angled triangle are defined as follows:
- cos(θ) = adjacent side/hypotenuse
- sin(θ) = opposite side/hypotenuse
For congruent angles, the cosine and sine values need not be the same, because cos(θ) and sin(θ) are generally different unless the angles are at specific measures.
The correct relationship between cosine and sine that is being hinted at in the question actually involves complementary angles, not congruent ones. Two angles are complementary if they add up to 90 degrees. For example, if angle X is 30 degrees, then its complement is 90 degrees - 30 degrees = 60 degrees.
Using the co-function identities in trigonometry, which state that the cosine of an angle is equal to the sine of its complement, we have:
cos(X) = sin(90 degrees - X)
Applying this to Sam's claim, we can say that cos X is equal to sin Y if and only if X and Y are complementary angles, meaning that X + Y = 90 degrees.
Options A, B, and D are not correct, because they propose that cos X sin Y if the angles are congruent, corresponding (which generally pertains to angles that occupy the same relative position at each intersection where a straight line crosses two others), or supplementary (which are angles that add up to 180 degrees), respectively.
Hence, the correct answer to this question is:
C. No, cos X = sin Y if X and Y are complementary angles.
