Let , 4−5 be a point on the terminal side of θ . find the exact values of cosθ , cscθ , and tanθ .

When point (4,-5) is plotted on a Cartesian plane, the point is located on the fourth quadrant. When you connect the point to the origin, the angle it forms with the +x axis, is denoted as θ. Now, we can use the pythagorean theorems. The identities for cosθ, cscθ and tanθ are:

cos θ = adjacent/hypotenuse = a/h
csc θ = hypotenuse/opposite = h/o
tan θ = opposite/adjacent = o/a

In a right triangle, there are three terms given to the sides with respect to the angle θ. The longest side is always the hypotenuse. The hypotenuse side is the distance from the origin (0,0) to point (4,-5). The distance is determined using the distance formula:

h = √(-5 - 0)² + (4-0)²
h = √41

With respect to the opposite side of θ is equal to 5 units, while the adjacent side is equal to 4 units. Using the identities mentioned above, their values are:

cos θ = 4/√41 = 0.625
csc θ = √41/5 = 1.281
tan θ = 5/4 = 1.25

The length of what is left would be 1 1/2 feet long.