Final answer:
To find sin a when tan a is 3/5, construct a right triangle with sides of length 3 and 5, and use the Pythagorean theorem to find the hypotenuse. Then use the definitions of sine and cosine based on a right triangle to calculate sin a and cos a.
Step-by-step explanation:
The problem at hand involves finding sin a when given that tan a is 3/5 for an acute angle a. To solve this, we can construct a right-angled triangle where the opposite side to angle a (which we can call Ay) has a length of 3 units, and the adjacent side (which we can call Ax) has a length of 5 units. According to the Pythagorean theorem, the hypotenuse (A) of this triangle can be calculated as A = √(3² + 5²) = √(9 + 25) = √34. Then, sin a is the ratio of the opposite side to the hypotenuse, so sin a = Ay / A = 3 / √34. To find the cosine, we use the ratio of the adjacent side to the hypotenuse, so cos a = Ax / A = 5 / √34. After simplifying, we get the final solution for sin a and cos a.