Final answer:
To find the trigonometric functions for t given tan t = 12/5 and 0 < t < π/2, we establish a right triangle with the sides of 12 and 5, derive the hypotenuse, and then calculate sin t, cos t, sec t, csc t, and cot t accordingly.
Step-by-step explanation:
If tan t = 12/5 and t is in the interval from 0 to π/2, we can use the Pythagorean identity to find the other trigonometric values for t. First, remember that the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. We'll consider a right triangle where the opposite side (to angle t) is 12, the adjacent side is 5, and we'll call the hypotenuse h.
From Pythagoras' theorem, h = √(12² + 5²) = √(144 + 25) = √169 = 13. Therefore, sin t = opposite/hypotenuse = 12/h = 12/13, and cos t = adjacent/hypotenuse = 5/h = 5/13.
Then:
- sec t = 1/cos t = h/adjacent = 13/5,
- csc t = 1/sin t = h/opposite = 13/12,
- cot t = 1/tan t = adjacent/opposite = 5/12.