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10.3 The Six Circular Functions and Fundamental Identities761In Exercises70-75, use Theorem10.10to answer the question. Assume thatθis an angle in aright triangle.70. Ifθ= 30◦and the side oppositeθhas length 4, how long is the side adjacent toθ?71. Ifθ= 15◦and the hypotenuse has length 10, how long is the side oppositeθ?72. Ifθ= 87◦and the side adjacent toθhas length 2, how long is the side oppositeθ?73. Ifθ= 38.2◦and the side oppositeθhas lengh 14, how long is the hypoteneuse?74. Ifθ= 2.05◦and the hypotenuse has length 3.98, how long is the side adjacent toθ?75. Ifθ= 42◦and the side adjacent toθhas length 31, how long is the side oppositeθ?76. A tree standing vertically on level ground casts a 120 foot long shadow. The angle of elevationfrom the end of the shadow to the top of the tree is 21.4◦. Find the height of the tree to thenearest foot. With the help of your classmates, research the termumbra versaand see whatit has to do with the shadow in this problem.77. The broadcast tower for radio station WSAZ (Home of “Algebra in the Morning with Carland Jeff”) has two enormous flashing red lights on it: one at the very top and one a fewfeet below the top. From a point 5000 feet away from the base of the tower on level groundthe angle of elevation to the top light is 7.970◦and to the second light is 7.125◦. Find thedistance between the lights to the nearest foot.78. On page753we defined the angle of inclination (also known as the angle of elevation) and inthis exercise we introduce a related angle - the angle of depression (also known as the angleof declination). The angle of depression of an object refers to the angle whose initial side isa horizontal line above the object and whose terminal side is the line-of-sight to the objectbelow the horizontal. This is represented schematically below.θhorizontalobserverobjectThe angle of depression from the horizontal to the object isθ(a) Show that if the horizontal is above and parallel to level ground then the angle ofdepression (from observer to object) and the angle of inclination (from object to observer)will be congruent because they are alternate interior angles.
762Foundations of Trigonometry(b) From a firetower 200 feet above level ground in the Sasquatch National Forest, a rangerspots a fire off in the distance. The angle of depression to the fire is 2.5◦. How far awayfrom the base of the tower is the fire?(c) The ranger in part78bsees a Sasquatch running directly from the fire towards thefiretower. The ranger takes two sightings. At the first sighting, the angle of depressionfrom the tower to the Sasquatch is 6◦. The second sighting, taken just 10 seconds later,gives the the angle of depression as 6.5◦. How far did the Saquatch travel in those 10seconds? Round your answer to the nearest foot. How fast is it running in miles perhour? Round your answer to the nearest mile per hour. If the Sasquatch keeps up this