What does a sine value of three-fifths indicate about the corresponding cosine in right triangle ABC?

In a right triangle, the sine and cosine values are related through the Pythagorean theorem. The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, a sine value of three-fifths means that if we consider the angle, the length of the side opposite the angle is 3 units, and the hypotenuse is 5 units.

To find the cosine of the angle, which is the ratio of the length of the adjacent side to the hypotenuse, we first need to determine the length of the adjacent side. By applying the Pythagorean theorem, we can find the length of the adjacent side (let's call it b):

a² + b² = c², where

• a is the opposite side (3),

• b is the adjacent side,

• c is the hypotenuse (5).

Substituting the known values, we get:

3² + b² = 5²

9 + b² = 25

b² = 25 - 9

b² = 16

b = 4.

Now we can calculate the cosine:

cosine = adjacent