In triangle J K L , cos K = 24 51 and angle J is a right angle. What is

Question

In triangle J K L , cos K = 24 51 and angle J is a right angle. What is the value of cos L ?

Accepted Answer

The correct answer is 15 17 . It's given that angle J is the right angle in triangle J K L . Therefore, the acute angles of triangle J K L are angle K and angle L . The hypotenuse of a right triangle is the side opposite its right angle. Therefore, the hypotenuse of triangle J K L is side K L . The cosine of an acute angle in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. It's given that cos K = 24 51 . This can be written as cos K = 8 17 . Since the cosine of angle K is a ratio, it follows that the length of the side adjacent to angle K is 8 n and the length of the hypotenuse is 17 n , where n is a constant. Therefore, J K = 8 n and K L = 17 n . The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. For triangle J K L , it follows that J K 2 + J L 2 = K L 2 . Substituting 8 n for J K and 17 n for K L yields 8 n 2 + J L 2 = 17 n 2 . This is equivalent to 64 n 2 + J L 2 = 289 n 2 . Subtracting 64 n 2 from each side of this equation yields J L 2 = 225 n 2 . Taking the square root of each side of this equation yields J L = 15 n . Since cos L = J L K L , it follows that cos L = 15 n 17 n , which can be rewritten as cos L = 15 17 . Note that 15/17, .8824, .8823, and 0.882 are examples of ways to enter a correct answer.

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