Find the exact values of sin alpha, cos alpha, tan alpha, sin beta, cos beta, and tan beta for right triangle. (4,8)

To find the exact values of the trigonometric functions for a right triangle with sides 4 and 8, we need to find the lengths of the remaining side and calculate the trigonometric ratios using those lengths.

Using the Pythagorean theorem, we can find the length of the hypotenuse (side opposite the right angle) as follows:

c^2 = a^2 + b^2
c^2 = 4^2 + 8^2
c^2 = 16 + 64
c^2 = 80
c = √80
c = 4√5

Now we can find the trigonometric ratios:

sin alpha = opposite/hypotenuse = 8/(4√5) = 2√5/√5 = 2
cos alpha = adjacent/hypotenuse = 4/(4√5) = 1/√5
tan alpha = opposite/adjacent = 8/4 = 2

sin beta = opposite/hypotenuse = 4/(4√5) = 1/√5
cos beta = adjacent/hypotenuse = 8/(4√5) = 2/√5
tan beta = opposite/adjacent = 4/8 = 1/2

Therefore, the exact values of the trigonometric functions for this right triangle are:

sin alpha = 2
cos alpha = 1/√5
tan alpha = 2
sin beta = 1/√5
cos beta = 2/√5
tan beta = 1/2