1. John built a zipline in his backyard. The maximum allowed slope for a zipline to be safe is 3.4°. The beginning of the zipline is 11 ft off the starting platform and the length of the zipline is 135.6ft. Does the zipline John built meet the safety requirements? Show your work and label the diagram to support your answer.

The maximum angle the zipline can have to be considered to be safe is 3.4 degrees, this represents the angle of elevation or the angle the zipline makes with the ground.

The vertical leg can represent the height of the beginning of the zipline and the hypotenuse can represent the zipline.

The problem asks whether the angle of the zipline is at most 3.4 degrees. Since we aren't given an explicit angle measure, we must use inverse trigonometric functions to solve for it!

Using Trigonometry

Inverse Functions

The inverse sine, cosine, tangent functions uses the appropriate side length ratios to find the angle, in our case in degrees, that makes the ratio.

Computing inverse functions are done with a calculator and most scientific calculators have all the inverse functions of the basic three trigonometric functions of sine, cosine, and tangent.

Note that none of the inverse functions can be written as [tex]\rm \dfrac{1}{sin(x)}[/tex], they are not the same. They usually look like this,

[tex]\rm sin^-^1(\dfrac{opp}{hyp} ), \: cos^-^1(\dfrac{adj}{hyp} ), \: tan^-^1(\dfrac{opp}{adj} )[/tex].

Applying Inverse Functions

Our created image informs us of the hypotenuse's length and the side length opposite to the angle as well, this fits the inverse sine function or, [tex]\rm sin^-^1(\dfrac{opp}{hyp})=x[/tex].

So,

[tex]\rm sin(\dfrac{11}{135.6})=4.65[/tex].

This tells us that the zipline's slope has a measure of 4.65 degrees, which is 1.25 more than the maximum allow angle, thus this zipline isn't safe.