PreCalc H Final Exam Review.pdf

Name:____________________________ Final Examination Review The final examination for this class will take place on ____________________. Calculators are allowed for the final examination. A reference sheet and a b...

Name:____________________________ Final Examination Review The final examination for this class will take place on ____________________. Calculators are allowed for the final examination. A reference sheet and a blank unit circle will be provided for the final examination. The final examination will cover: Trig Unit 1 Trig Unit 2 Trig Unit 3 Polars & Parametrics Vectors Sequences & Sums The format of the final examination will be: 20 MC questions @ 2 pts each (40 pts) 10 Short Answer questions @ 6 pts each (60 pts) 1.) Eliminate the parameter and write the corresponding rectangular equation: 𝑥 = 2𝑡 2 𝑦 =𝑡 2.) Are the vectors 𝑢 = 3𝑖 + 4𝑗 and 𝑣 =− 2𝑖 + 3𝑗 orthogonal? Provide both algebraic and written explanations. PreCalculus Honors Mr. Lanoie  𝑡ℎ 3.) Find a formula for 𝑎𝑛 for the arithmetic sequence given. Then, find the 30 term in the sequence. 𝑎1 = 5 , 𝑎4 = 15 4.) Find all solutions of the equation in the interval [0, 2π). π π 𝑐𝑜𝑠(𝑥 + 4 ) − 𝑐𝑜𝑠(𝑥 − 4 ) =1 5.) Sketch the graph of the provided function on the coordinate axis. Include at least two cycles on your graph. 𝑦 =− 2𝑐𝑜𝑠(2π𝑥) + 1 PreCalculus Honors Mr. Lanoie  6.) The distance from Chicago to St. Louis is 440 kilometers, from St. Louis to Atlanta is 795 kilometers, and from Atlanta to Chicago is 950 kilometers. What are the angles in the triangle with these three cities as vertices? 7.) Sketch the following polar curves. Then, algebraically find their point(s) of intersection. Put the angle(s) in radian measure. 𝑟 = 2 + 2𝑐𝑜𝑠θ 𝑟=3 8.) Solve the equation for 0 ≤ 𝑥 < 2π. 2 4𝑐𝑜𝑠 𝑥 − 3 = 0 PreCalculus Honors Mr. Lanoie  9.) The number of hours of daylight throughout the year is related to the periodic change in the seasons. The corresponding change in the tilt of the earth also causes the number of daylight hours to change. Because such changes are periodic from year to year, the number of daylight hours on a given day is (nearly) the same every year, with each year (365 days) representing one full sinusoidal cycle. This sinusoidal graph will be different for every location on earth, varying by latitude. Boston, MA is at latitude 42. 3°. At this location, the shortest day has approximately 9.1355 hours of daylight. The longest day, which occurs exactly one half of a 365-day year later, has approximately 15.2345 hours of daylight. Create a graph and include a trigonometric equation to represent the yearly number of daylight hours in Boston, MA. 11π 10.) Use a half angle formula to find the exact value of 𝑐𝑜𝑠( 12 ). PreCalculus Honors Mr. Lanoie  11.) A plane’s heading is 160° and its air speed is 350 mph. If a wind is blowing east at 20 mph, what are the plane’s ground speed and true course? 𝑡ℎ 12.) a.) Find the 𝑛 term of the geometric sequence. 3 3 16 , 4 , 3,......, 𝑛 = 12 𝑡ℎ b.) Find the 20 partial sum of the sequence. 13.) What is the domain of the function 𝑦 = 𝑡𝑎𝑛𝑥? PreCalculus Honors Mr. Lanoie  π 14.) Find three additional polar representations of the point (5, − 3 ) on − 2π < θ < 2π. 15.) Find the corresponding rectangular coordinates for the polar coordinates 7π (2, 6 ). 16.) Provide an equation for each of the following and sketch a rough graph of the equation’s graphical representation. a.) a dimpled limaçon: b.) a cardioid: c.) an inner-loop limaçon: d.) a convex limaçon: PreCalculus Honors Mr. Lanoie  17.) If 𝑢 = 𝑎𝑛𝑑 𝑣 =< 1, − 3 >, a.) find 2𝑢 − 3𝑣. b.)find 𝑢 𝑣 − ‖𝑢‖. 18.) If 240° represents a bearing, what would the measure of a related direction angle θ be? 19.) Find the angle between vectors 𝑢 =< 4, − 3 > and 𝑣 =< 1, 2 >. 20.) Find the magnitude and direction angle of the vector 3𝑖 − 4𝑗. PreCalculus Honors Mr. Lanoie  21.) Find a unit vector in the direction of the vector 𝑤 =− 3𝑖 − 5𝑗. 22.) Find 𝑢 − 𝑣 if 𝑢 has a magnitude of 85 and a direction of 45° and 𝑣 has a magnitude of 100 and a direction of 327°. 23.) Identify the transformations given the function 𝑦 =− 5𝑠𝑖𝑛(4𝑥 + π) − 3. 24.) If you wanted to sketch a graph of the cosecant function, it might be helpful to first sketch which function? 25.) Provide a rough sketch of the secant function below. PreCalculus Honors Mr. Lanoie  26.) Solve △𝐴𝐵𝐶 under the given conditions: ∠𝐴 = 92°, ∠𝐵 = 28°, and 𝑎 = 15. 27.) Find the values of 𝑥, in radians, on the interval 0 ≤ 𝑥 < 2π that make 𝑐𝑜𝑠2𝑥 + 𝑐𝑜𝑠𝑥 = 0 true. −1 1 28.) Determine the value of 𝑐𝑜𝑠(𝑠𝑖𝑛 (− 2 )). PreCalculus Honors Mr. Lanoie  −15 3π 4 π 29.) Let 𝑠𝑖𝑛(𝑥) = 17 where 2 < 𝑥 < 2π and 𝑐𝑜𝑠(𝑦) = 5 where 0 < 𝑦 < 2. Find 𝑐𝑜𝑠(𝑥 − 𝑦). 2 30.) What is the expression 𝑐𝑜𝑠(2𝑥) + 𝑠𝑖𝑛 (𝑥) equivalent to? PreCalculus Honors Mr. Lanoie