Question

The point `(3 1/2, -7 3/4)` is on the terminal side of an angle in standard position, how do you determine the exact values of the six trigonometric functions of the angle?

Answer 1

The point `(x = 7/2, y = - 31/4)`, on the terminal side of the angle (or arc) t, lies in Quadrant 4.
`tan t = y/x = 7/2 : -31/4 = (7/2)(-4/31) = - 14/31`
`cos^2 t = 1/(1 + tan^2 t) = 1/(1 + 196/961) = 961/1157`
`cos t = 31/sqrt1157` (because t lies in Quadrant 4)
`sin t = cos t.tan t = (31/sqrt1157)(- 14/31) = - 14/sqrt1157`
`cot t = 1/(tan) = - 31/14`
`sec t = 1/(cos) = sqrt1157/31`
`csc t = 1/(sin) = - sqrt1157/14`