A right triangle has acute angles C and D. If cot C= 8/15 and cos D = 15/17, what are tan D and sin C?
To find tan D, we can use the formula:
tan D = sin D / cos D
First, we need to find sin D. Since cos D = 15/17, we can use the Pythagorean identity sin^2 D + cos^2 D = 1 to find sin D:
sin^2 D + (15/17)^2 = 1
sin^2 D + 225/289 = 1
sin^2 D = 1 - 225/289
sin^2 D = 64/289
sin D = sqrt(64/289)
sin D = 8/17
Now, we can find tan D:
tan D = sin D / cos D
tan D = (8/17) / (15/17)
tan D = 8/15
Therefore, tan D = 8/15.
To find sin C, we can use the formula:
sin^2 C + cos^2 C = 1
Since cot C = 8/15, we know that cot C = cos C / sin C. We can use this information to find sin C.
cot C = 8/15
cos C / sin C = 8/15
cos C = 15/17
Now we can find sin C:
sin^2 C + (15/17)^2 = 1
sin^2 C + 225/289 = 1
sin^2 C = 1 - 225/289
sin^2 C = 64/289
sin C = sqrt(64/289)
sin C = 8/17
Therefore, sin C = 8/17.