# Do this as follows a carefully sketch the auxiliary

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Sketch the given curve in polar coordinates. Do this as follows: (a) Carefully sketch the auxiliary curve, a rectangular graph on the θ ,r coordinate system provided. (b) Then translate this graph to the polar one, the x,y one. Equation: r = 1 + 2cos( θ ) (a) r θ (b) Polar y x 10. (5 pts.) Write cos(cos -1 (u ) + sin -1 (v)) as an algebraic expression containing u and v. cos(cos -1 (u ) + sin -1 (v)) = 11. (5 pts.) Find the exact value of cos -1 (cos(-7 π /5)). cos -1 (cos(-7 π /5)) =

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Final Exam/MAC1114 Page 7 of 8 12. (10 pts.) To measure the height of the top of a distant object , a surveyor takes two sightings of the top of the object 100 feet apart. The first sighting, which is nearest the object, results in an angle of elevation of 45°. The second sighting, which is most distant from the object, results in an angle of elevation of 30°. If the transit used to make the sightings is 5 feet tall, what is the height of the object. You may assume the object is on a level plane with the base of the transit. 13. (5 pts.) Find the exact value of tan(2 tan -1 (1/4)). tan(2 tan -1 (1/4)) = 14. (5 pts.) Using DeMoivre’s Theorem, find all the complex cube roots of -i. [Don’t confuse cube roots with cubes!!] 15. (5 pts.) Two sides of a triangle have lengths 3 feet and 4 feet, and the angle included between the two sides is 15°. What is the exact length of the third side?
Final Exam/MAC1114 Page 8 of 8 16. (10 pts.) Establish the following identity. [ Hint: Begin on the left side, expand and clean using appropriate identities, as you work towards the right. Be very patient. ] 2sin(( α + β )/2)cos(( α - β )/2) = sin( α ) + sin( β ) 17. (5 pts.) Obtain the exact value of sin(9 π /8). Show clearly and neatly all the uses of appropriate identities. sin(9 π /8) = 18. (5 pts.) Express the following product as a sum containing only sines or cosines. cos(5 θ )cos(3 θ ) = 19. (5 pts.) Use the Law of Sines to solve the triangle with α = 105°, γ = 35°, and c = 24. You may assume that the standard labelling scheme is used.
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