Mathematics

Mathematics

Problem

In △ABC\triangle ABC, ∠B=90∘\angle B = 90^\circ. Two students make the following observations: • Student X measures the ratio of the side opposite ∠A\angle A to the hypotenuse and obtains the value 12\dfrac12. • Student Y measures the ratio of the side adjacent to ∠C\angle C to the hypotenuse and also obtains 12\dfrac12. a. Using the observations, determine the measures of ∠A\angle A and ∠C\angle C. b. Show that the two acute angles of the triangle are complementary. c. If the hypotenuse is 10 cm, calculate the lengths of all the sides of the triangle.
Intuitive explanation

Both students are using the fundamental trigonometric ratios in a right triangle, but they're measuring different angles. The key insight is recognizing that ∠A\angle A and ∠C\angle C are complementary (they sum to 90°90°), so the "opposite to ∠A\angle A" and "adjacent to ∠C\angle C" actually refer to the same side. When both ratios equal 12\frac{1}{2}, you're really getting one constraint that determines the triangle's angles—specifically, which special angle has a sine or cosine of 12\frac{1}{2}.

Intuitive explanation

Both students are using the fundamental trigonometric ratios in a right triangle, but they're measuring different angles. The key insight is recognizing that ∠A\angle A and ∠C\angle C are complementary (they sum to 90°90°), so the "opposite to ∠A\angle A" and "adjacent to ∠C\angle C" actually refer to the same side. When both ratios equal 12\frac{1}{2}, you're really getting one constraint that determines the triangle's angles—specifically, which special angle has a sine or cosine of 12\frac{1}{2}.

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