Final answer:
The measures of the two acute angles were determined using the complementary angle property and the given ratio between the angles. However, the calculated angle measures do not match any of the provided options, suggesting there may be an error in the question or the answer choices.
Step-by-step explanation:
The question asks us to find the measures of two acute angles of a right triangle where the sine of one angle is equal to the cosine of the other angle, and the measure of the second angle is two-thirds the measure of the first angle. We will use the complementary angle property, which states that in a right triangle, the sine of one acute angle is equal to the cosine of the complementary angle, i.e., sin(x) = cos(90° - x).
Let's denote the measures of angle 1 and angle 2 as A and B, respectively. According to the problem, we have sin(A) = cos(B) and B = (2/3)A. From the complementary angle property, we can deduce that cos(B) = sin(90° - B). Therefore, sin(A) = sin(90° - B). Since sin is a one-to-one function within the range of acute angles, A must equal 90° - B.
Setting up the equation A + B = 90° (since the angles are complementary in a right triangle), and substituting B with (2/3)A, we obtain A + (2/3)A = 90°, which simplifies to (5/3)A = 90°, and further A = (3/5) * 90° = 54°. We use the earlier equation B = (2/3)A to find that B = (2/3) * 54° = 36°. This scenario is not explicitly listed in the provided options, implying a possible mistake in the question or the options.