Math 112 Recitation Assignment Name: _________________________________ Week 10

Final Exam Review

NOTE: These problems are only for reviewing the material covered since the second midterm. To review the concepts covered earlier in the term, it will be a good idea to study your recitation assignments from Week 5 and Week 8 (the midterm reviews) AND your midterm exams.

1. (A) Determine if AAS, ASA, SSA, SAS, or SSS is given.

(B) Decide if the Law of Sines of the Law of Cosines should be used first to solve the triangle.

(C) Solve the triangle, if possible. If the triangle represents the ambiguous case, solve for all possible triangles. Approximate values to the nearest tenth.

a. 𝑎 = 45, 𝑏 = 24, and 𝛾 = 35° b. 𝑎 = 5, 𝑏 = 3, and 𝛽 = 52° c. 𝛼 = 55.2°, 𝛾 = 114.8°, 𝑏 = 19.5 d. 𝑎 = 7, 𝑏 = 9, and 𝛼 = 20° e. 𝑎 = 4.2, 𝑏 = 5.1, and 𝑐 = 3.7

2. Approximate the area of each triangle to the nearest tenth.

a. 𝑎 = 12.3, 𝑏 = 13.7, and 𝛾 = 39° b. 𝑎 = 2.1, 𝑏 = 1.7, and 𝑐 = 2.2

3. To approximate the length of a lake, a surveyor starts at one end of the lake and walks 245 yards. He then turns 110º and walks 270 yards until he arrives at the other end of the lake. Approximately how long is the lake?

Final exam information:

When is the final?______________________________________________________________________

Where is the final? _____________________________________________________________________

What should you take with you to the final? _________________________________________________

Working through these problems (and discussing them with your classmates and TA) will be helpful in reviewing for the final exam. However, they should not be thought of as a guarantee of what will (or will not) be asked on the exam. Working through these problems should only be one component of your preparation for the exam. Reviewing lecture notes, recitation assignments and homework will also be important.

4. Two ships leave port at 4pm. One is headed at a bearing of N 38 E and is traveling at 11.5 miles per hour. The other is traveling 13 miles per hour at a bearing of S 47 E. How far apart are they at 6pm?

5. Two airplanes leave an airport, and the angle between their flight paths is 40º. An hour later, one plane has traveled 300 miles while the other has traveled 200 miles. How far apart are the planes at this time?

6. Sketch a vector v that models a 20-mph wind blowing toward the north. What are the

horizontal and vertical components of v? Find 2v and − !!𝐯. How would you interpret each

of these results? 7. Find the magnitude and direction angle 𝜃 for the vector −1, 3 , where 0° ≤ 𝜃 < 360°. 8. A vector has magnitude 9 and direction angle 225°. Write the horizontal and vertical

components for the vector. Give an exact answer AND round to the nearest tenth. 9. A vector has initial point 𝑃 = (−1,−2) and terminal point 𝑄 = (4,4). Write the vector as

𝐯 = 𝑥, 𝑦 and find the magnitude of v. 10. Use the figure to evaluate:

a. u + v b. 2u – v c. –u d. 𝐯

11. Use the parallelogram rule to find the magnitude of the resultant force for the two forces

shown in the figure below.

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

u

v

65°  

102  lb.  

85  lb.  

12. A plane flies on a bearing of 230° at 350 miles per hour. A wind is blowing from the west at

30 miles per hour.

a. Find vectors v and u that model the velocity of the airplane and the velocity of the wind, respectively.

b. Use vectors to determine the groundspeed of the plane. c. Find the final bearing of the plane in the wind.

13. Find the exact value of (A) cos 2𝜃 and (B) 𝑐os !!, given that 𝑐os𝜃 = − !

! and 𝜃 is in quadrant

II. In which quadrants are the angles 2𝜃 and !!?

14. Use a half-angle identity to find the exact value of sin  (−67.5°). 15. Find all solutions, expressed in radians: cos 2𝑡 = sin 𝑡.

Remember to study more stuff, too J