Math 112 Week 6 The Law of Cosines and Sines When to Use the Laws of Cosines and Sines • When two sides of a triangle and the angle between them are known the Law of Cosines is useful. It is also useful if all three sides of a triangle are known. 1.An aerial tram starts at a point one half mile from the base of a mountain whose face has a pi/3 angle of elevation. The tram ascends at an angle of pi/9. How far does the tram travel? 2. A post is supported by two wires(one on each side going in opposite directions) creating an angle of 80 degrees between the wires. The end of the wires are 12m apart onthe ground with one wire forming an angle of 40 degrees with the ground. Find the lengths of the wires. 3.Two scuba divers are 20 meters apart just below the surface of the water. They both spot a shark below them and between them. The angle of depression from the first diver to the shark is 0.82. The angle of depression from the second diver to the shark is 0.698. How far are each of the divers from the shark?How far below the surface of the water is the shark?• The Law of Sines is useful when we know a side and the angle opposite it and one other angle or one other side. 1.A person leaves her home and walks 6 miles due east and then 3 miles northeast. a)How far has she walked? b)How far away from home is she? c)At what angle must she walk to go directly home? 2.Two ships are sailing from Halifax. The Nina is sailing due east and the Pinta is sailing 0.75(radians) south of east. After an hour, the Nina has travelled 115 km and the Pinta has traveled 98 km. How far apart are the two ships? 3.Click on the graph to view a larger graph Use the Law of Cosines to find the indicated angle x given in the graph4. Two ships leave a harbor at the same time, traveling on courses that have an angle of 120 degrees between them. If the first ship travels at 30 miles per hour and the second ship travels at 25 miles per hour, how far apart are the two ships after 1.5 hours?5. A pilot flies in a straight path for 1 hour and 30 min. She then makes a course correction, heading 10 degrees to the right of her original course, and flies 2 hours in the new direction. If she maintains a constant speed of 695 miles per hour, how far is she from her starting position?6. A surveyor determines that the angle of elevation to the top of a building from a point on the ground is 26.7 degrees. He then moves back 57.9 feet and determines that the angle of elevation is 20.6 degrees. What is the height of the building? Round your answer to four decimal places. Trigonometric Functions 1. Using your calculator, find, (if possible), a solution for θ in radians. If no solution exists, enter NONE. tan(θ−4)=0.142.Find all solutions to the equation below for 0≤ ≤α π2 . If there is more than one answer, enter your solutions in a commaseparated list, and be sure to give exact answers without any rounding. 2cos( )=1 α 3.Find all solutions to the equation with 0≤ ≤α π2 . Give an exact answer if possible, otherwise give value(s) of αα accurate to at least four decimal places. 4tan( )+9=6 α4.If possible, find a solution to sin(θ+5)=−45. If no solution exists, enter NONE. 5. Find all solutions of the equation 2cos3x=1 in the interval [0,π). Special Angles 1.One solution to the equation below is theta=arccos(0.3)=72.54 degrees. Find a different solution between 0 degree and 360 degrees. Graphs of Sine and Cosine The Unit Circle and the Sine and Cosine Functions The Sine and Cosine functions in Relation to Right Triangles Modeling with Sine and CosineThe tangent Function Inverse Trigonometric Functions Radians and Arc Length Basic Geometry The Sine and Cosine Functions 1. Find a plausible formula for the function r(s) described by the following graph.2.Find an equation for the line L(x) in the picture below.