MATH 101 AP Calculus AB - Module 10, Using Trig Functions Assignment.

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Module 10, Using Trig Functions Assignment This summer you and your friend Mikhail are going to search for sunken treasure with a professional team of divers. You will help the team locate likely areas to search for the items you’ve been hired to find, plan out expeditions, and you will also travel with the team to carry out plans.

Although there are many missions to complete, one specific item you have been hired to find is called The Cylinder of Fate. The cylinder is jewel-encrusted and supposedly it will bring the owner good luck in all aspects of life.

According the legend, this treasure was lost when a pirate ship named The Howler sank in rough seas off the coast of a local island.

You have read all the material you could find about The Howler and about The Cylinder of Fate. Based on this reading and some information about the sea floor and tides in the area where The Howler was thought to have sunk, you suggest that the team start by taking the search boat 65 meters due east of shore. At this distance the angle of depression between the shore and the hypothetical location of The Howler and its treasure should be about 30 ° . The search boat will be at the vertex of a 90 ° angle between the shore and the treasure below.

Use this information and what you know about solving triangles using trigonometric functions to explore the questions below.

1. Draw a picture of the scenario described. Label the location of the shore, the search boat, the treasure, as well as the angles and lengths.

Boat

65 meters

Shore 30 degrees

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Treasure 30 degress

2. What is the approximate vertical distance between the search boat and the treasure? You can round to the nearest hundredth, as needed. Show your work.

37.53 is the distance from the boat to the treasure Tan30=y/65

3. What is the straight-line distance between the treasure and the shore? You can round to the nearest hundredth, as needed. Show your work.

2

2

C= 65 +¿ 37.53 √¿ C= 75.06m

4. Use inverse trigonometric functions to show that the measure of the angle that the treasure creates with search boat and the shore is about 1.05 radians. Show your work and explain the steps you used to solve.

Sin x = 65/75.06 Sin -1(65/75.06) =x=60degrees x π/180 = 1.05 radians

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When your team sends the divers down at the location you recommended, they find the remnants of a ship, but there is no treasure aboard. Based on the photographs the divers took, you and the team think the boat is not The Howler, but another pirate ship called The Serpent Princess. This ship was known to travel with The Howler, so the team suspects you have found the right general area to search.

After some additional reading, you think that The Serpent Princess and The Howler may have sank during the same storm. Based on the sizes of each boat, the two wrecks would need to be about 50 meters apart on the sea floor. Your captain maneuvers the search boat into place. She activates the sonar and sees there is something that resembles a ship about 18 meters directly below the search boat.

5. Draw a picture of this scenario. Label the location of The Howler, The Serpent Princess, and the search boat, as well as the angles and lengths.

Search Boat

18

The serpent princess]

50 Howler meters 6. Find the degree measure of the angle of elevation between the location of The Serpent Princess and the search boat. You can round to the nearest hundredth, as needed. Show your work.

X=19.8 degrees Tan x=18/50 Tan (18/50)=x=19.8

7. Solve for the rest of the missing side lengths and angles in the right triangle that represents this scenario. Show your work and explain what each of these numbers represents in this context.

You can round to the nearest hundredth, as needed.

C= √ 502+ 182 C=53.14 meters 180-90-19.8=70.2degress

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