Suppose that è is an angle in standard position whose terminal side intersects the unit circle at ((-15/13),(12/13)).

Find the exact values of cscè, tanè, and sinè.

A terminal position of ((-15/13),(12/13))

puts you in quadrant II and using similar triangles we could have the same angle for a terminal position at
(-5,4)
so r = √(25+16) = √41

x = -5, y = +4 , r = √41

sin è = 4/√41 ----> csc è = √41/4
tan è = =4/5

To find the exact values of cscè, tanè, and sinè, we need to use the given coordinates to determine the values of the trigonometric functions.

First, let's identify the values of the legs of the right triangle formed by the given coordinates and the origin (0, 0).

The x-coordinate, -15/13, represents the adjacent side, and the y-coordinate, 12/13, represents the opposite side. The hypotenuse of the triangle, which is the radius of the unit circle, is always 1.

Using the Pythagorean theorem, we can find the length of the hypotenuse:

hypotenuse = √(adjacent^2 + opposite^2)
= √((-15/13)^2 + (12/13)^2)
= √(225/169 + 144/169)
= √(369/169)
= √369 / 13

Now, we can calculate the values of the trigonometric functions:

cscè = 1/sinè
= 1/(opposite/hypotenuse)
= hypotenuse/opposite
= (√369 / 13) / (12/13)
= √369 / 12

tanè = opposite/adjacent
= (12/13) / (-15/13)
= 12/-15
= -4/5

sinè = opposite/hypotenuse
= (12/13) / (√369 / 13)
= 12/√369

Therefore, the exact values of cscè, tanè, and sinè are:

cscè = √369 / 12
tanè = -4/5
sinè = 12/√369