Precalculus 2e

Chapter 7 Group Project

The Law of Sines and Law of Cosines

Primary Name: _______________________________             Team Member: _______________________________

Team Member: _______________________________               Team Member: _______________________________

The real-world application of trigonometry often requires that many different elements of trig work together to find a solution. These may include basic trigonometric facts (the angles in a triangle sum to 180º, vertical angles are equal, and so on), special values, the use of identities, multiple applications of trigonometric properties, and other elements. In this project we will use a combination of the law of sines and the law of cosines so solve practical applications.

     I.    Project Survey: A surveyor at point A on the south side of a canyon needs to find the distance between the tree T and the rock formation R on the canyon’s north side (, see figure). To make this possible, she takes a bearing from point A, and finds that point R is 70º East of North, and point T is 30º East of North. She then measures a straight line distance of 30 m to a point D on her side of the canyon. From there she finds that point T is 68º West of North, and point R is 9º West of North. Assume the dashed lines intersect at a point C. Use this information to answer the questions that follow.

          1)    What is the measure of ÐRAD?                                                 2)    What is the measure of ÐTAR?  

          3)    What is the measure of ÐTDA?                                                  4)    What is the measure of ÐRDT?  

          5)    What is the measure of ÐACD? Why?                                      6)    What is the measure of ÐTCR? Why?

          7)    What is the measure of ÐTCA? Why?                                      8)    What is the measure of ÐRCD? Why?

          9)    Find the measure of side .                                                  10)    Find the measure of side .

        11)    What is the measure of ÐCTA? Why?                                    12)    What is the measure of ÐCRD? Why?

        13)    Find the measure of side .                                                  14)    Find the measure of side .

        15)    Find the distance  between the tree and                           16)    Does this answer (to Exercise 15) seem reasonable?

                  the rock formation.

    II.    Project Rescue: The Park Rangers protecting our national forests typically carry communication and sighting equipment that is used to help locate lost hikers, forest fires, airplanes crashes and other such like. In one instance, a light plane P crashed in a dense part of the forest, with the crash occurring to the northeast of a ranger R who witnessed the event. The witness was located 12 miles due east of the ranger station S. The sighting from the witness to the crash site was 20º East of North, while the sighting from the Ranger Station, based on the smoke plume from the wreckage, was 52º East of North.

        17)    Use the information given to draw rRSP. Draw the triangle large enough to fit the space below, then label

        18)    What is the measure of ÐPSR? Why?

        19)    What is the measure of ÐPRS? Why?

        20)    What is the measure of ÐRPS? Why?                                     21)    Can we use the Law of Sines to find the

                                                                                                                                    distances  and ? Why or why not?

        20)    What is the distance  from the ranger                               21)    What is the distance  from the ranger who

                  station to the crash site?                                                                       witnessed the crash to the crash site?

        22)    A search and rescue team T is assembling at a point that is 7 miles due east of the ranger station S, directly between

                  the ranger station and the witness. Redraw the figure to include a line from point T to point P.

                  (a) How far is the rescue team T from the crash site?

         (b)    What direction should they head to reach the

                  crash site? Answer in terms of a bearing.

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