To find the values of the trigonometric functions, we need to determine the ratios for the given tangent value (tan(t)) and restrict the angle t to the given interval (T < t < 2π/31).
Given: tan(t) = 5/12
From this, we can determine the values of the other trigonometric functions:
sin(t) = (5/12) / sqrt(1 + (5/12)^2)
cos(t) = 1 / sqrt(1 + (5/12)^2)
sec(t) = 1 / cos(t)
csc(t) = 1 / sin(t)
cot(t) = 1 / tan(t)
Let's calculate these values:
sin(t) = (5/12) / sqrt(1 + (5/12)^2) = (5/12) / sqrt(1 + 25/144) = (5/12) / sqrt(169/144) = (5/12) / (13/12) = 5/13
cos(t) = 1 / sqrt(1 + (5/12)^2) = 1 / sqrt(1 + 25/144) = 1 / sqrt(169/144) = 1 / (13/12) = 12/13
sec(t) = 1 / cos(t) = 1 / (12/13) = 13/12
csc(t) = 1 / sin(t) = 1 / (5/13) = 13/5
cot(t) = 1 / tan(t) = 1 / (5/12) = 12/5
Therefore, the exact values of the trigonometric functions within the given interval are:
sin(t) = 5/13
cos(t) = 12/13
sec(t) = 13/12
csc(t) = 13/5
cot(t) = 12/5