Answer:
In a right triangle, one angle is always 90 degrees (a right angle). The other two angles are acute angles, which means they are less than 90 degrees.
Let's call the acute angle we're interested in "a". We know that sin a = 3/5.
"Sin" is short for "sine", which is a ratio of two sides of the triangle. Specifically, it's the ratio of the length of the side opposite angle a to the length of the hypotenuse (the longest side of the triangle, which is always opposite the right angle).
So, in our triangle, if sin a = 3/5, that means the side opposite angle a is 3 units long and the hypotenuse is 5 units long.
Now, we can use the Pythagorean theorem to find the length of the third side of the triangle (the one adjacent to angle a). The Pythagorean theorem says that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In other words:
a^2 + b^2 = c^2
where a and b are the lengths of the two shorter sides, and c is the length of the hypotenuse.
In our triangle, we know that b is the side adjacent to angle a, so we're trying to find its length. We also know that a = 3 and c = 5, so we can plug those values into the Pythagorean theorem and solve for b:
3^2 + b^2 = 5^2
9 + b^2 = 25
b^2 = 16
b = 4
So the length of the side adjacent to angle a is 4.
Now, we can use the ratios of the trigonometric functions (sine, cosine, and tangent) to find the values of cosine and tangent for angle a.
Cosine (cos) is the ratio of the length of the adjacent side to the length of the hypotenuse. So in our triangle:
cos a = adjacent/hypotenuse = 4/5
Tangent (tan) is the ratio of the length of the opposite side to the length of the adjacent side. So in our triangle:
tan a = opposite/adjacent = 3/4
Therefore, if sin a = 3/5 in a right triangle, where a is an acute angle, then cos a = 4/5 and tan a = 3/4.