Question

Question 1. Part 1: Use the trigonometric ratios sine, cosine, and tangent to answer the following questions. Find the sine, cosine, and tangent ratios of angle X.

Part 2: Use the trigonometric ratios sine, cosine, and tangent to answer the following questions. Find the sine, cosine, and tangent ratios of angle Y.

Question 2. A ski resort is building a new ski lift that will transport tourists from the base of the mountain to its highest point. This mountain has a vertical height of 200 yards, and the ski lift will rise at an angle of 40 degrees. When the project is completed, how many yards, d, will a tourist travel from the base of the mountain to its peak?

Part I: Sketch a figure to illustrate the scenario above. Label the vertices and the lengths that are given in the question.

Part II: Using your sketch from Part I, write an equation using a trigonometric ratio to find the distance a tourist will travel from the base of the mountain to its peak. Round your answer to the nearest 100th. Show your work.

Answer 1

1.sin x = \frac{5}{13}

cos x = \frac{12}{13}

tan x = \frac{5}{12}

sin y = \frac{12}{13}

cos y = \frac{5}{13}

tan y = \frac{12}{5}

2. ditance travelled = hypotenuse = \frac{200}{sin 40} = 311.14yards

Explanation:

sin x =
`(height)/(hypotenuse)`

cos x =
`(base)/(hypotenuse)`

tan x =
`(height)/(base)`

therefore from the figure sin x = \frac{5}{13}

cos x = \frac{12}{13}

tan x = \frac{5}{12}

according to angle y height will be 12 and base will be 5

therefore sin y = \frac{12}{13}

cos y = \frac{5}{13}

tan y = \frac{12}{5}

2. given that height = 200 yards and angle x= 40

we know that tan x = \frac{height}{base}

therefore tan 40 = \frac{200}{base}

therefore base = \frac{200}{tan 40}

they have asked us to find hypotenuse therefore

sin x = \frac{height}{hypotenuse}

hypotenuse = \frac{200}{sin 40} = 311.14yards