Math 113 Fall 2023 Final Exam (PDF)

Department of Mathematics, University of Wisconsin-Madison Math 113 Fall 2023 Final Exam Name : Solutions Campus ID : Bubble in the circle for your section below: Section 001 TR 8:00 - 9:15...

Department of Mathematics, University of Wisconsin-Madison Math 113 Fall 2023 Final Exam Name : Solutions Campus ID : Bubble in the circle for your section below: Section 001 TR 8:00 - 9:15 Karen Duffy Section 002 TR 1:00 - 2:15 John Yin Section 003 TR 11:00 - 12:15 Ian Seong Section 004 TR 1:00 - 2:15 Eugenia Malitsky Section 005 TR 2:30 - 3:45 Eugenia Malitsky Section 006 TR 4:00 - 5:15 Eugenia Malitsky Directions: Work on the following problems, showing all of your work, because if there is NO WORK, then there is NO CREDIT! When asked to give explanations, make sure your explanations are clear and detailed. Also, you DO NOT have to rationalize expres- sions, so please don’t! There are two sections to this exam: Multiple-Choice and Written. The multiple-choice problems count for a total of 20 points while the written problems count for a total of 80 points. You may not use textbooks, course notes, cell phones, or cal- culators during this exam. Again, you are required to show your work on each problem on this exam, and the following rules apply: Organize your work in a reasonable, neat, and coherent way. Work that is jumbled, disorganized, and lacks clear reasoning will receive little or no credit. Unsupported answers will not receive full credit. Any answer must be supported by calculations, explanation, and/or algebraic work to receive full credit. Partial credit may be given to well-argued, incorrect/partial solutions also. If you require more space, use the extra stapled pages. Clearly indicate when you do this below the problem. Do NOT un-staple the pages! In the multiple choice section do NOT make stray marks inside the boxed area: bubble in a single answer and do NOT mark the other choices in ANY way. In the written section, please write ONLY inside the BOXED area, i.e. do NOT write outside the box. 1. (1 pt) Find the complement of the angle p4. a. p b. p c. p 4 complementary angles add to E 90 4 d. p 3 e. 4 p I I E 2. (1 pt) Find the radian measure of the central angle of a circle of radius r = 3 inches and arc length s = 21 inches. a. q = 63 rad b. q = 7p rad s ro c. q = 1 7 rad 21 30 d. q = 7 0 p 7 rad e. q = 7 rad 3. (1 pt) Find the value of cos( 11p 6 ). 1 a. p 2 I 3 b. 2 c. 0 1 d. 2 p 3 e. I G 2 6 L Cos F 4. (1 pt) Find the value of sin(t + p2 ) if cos(t) = 56. 5 a. 6 b. 5 6 Sin t E sin t cos E cos t sin E c. Sin t o E i 6 d. 5 6 e. 5 p 5. (2 pts) At a certain time of the day a light post, 5m tall, has a shadow of 5 3m. Calculate the value of q, the angle of elevation of the sun at that time. I a. 30 b. 60 I lightpost tanto c. 45 d. 90 5m r e. 0 0 30 6. (2 pts) The terminal side of q lies on the line 3x 4y = 0 in the third quadrant. Find the exact value of cot q. 4 a. 3 3x 4g 0 y 3 b. 4 c. 4 5 choose 4 Myka y.tt d. 5 4 Then y C4 3 kegaf.ve k 4 o in Q3 e. 3 Cotto  Period 7. (2 pts) Consider the parent function f ( x ) = cos x. Describe the sequence of trans- formations from f to the function g given by g( x ) = cos(2x p ) + 3. cos 2 Effy up a. The graph of g( x ) contains exactly one period in the interval [p, 2p ] and g( x ) is obtained by shifting f ( x ) downward 3 units. in widtion ⇥ ⇤ b. The graph of g( x ) contains exactly one period in the interval p2 , 3p2 and g( x ) is obtained by shifting f ( x ) upward 3 units. ⇥ ⇤ c. The graph of f ( x ) contains exactly one period in the interval p2 , 3p 2 and g( x ) is obtained by shifting f ( x ) upward 2 units. wrongsize ⇥ ofshift⇤ d. The graph of g( x ) contains exactly one period in the interval p2 , 3p2 and g( x ) is obtained by shifting f ( x ) downward p2 units. ton kids e. The graph of g( x ) contains exactly one period in the interval [p, 3p ] and g( x ) is obtained by shifting f ( x ) upward 3 units. period Trong 8. (2 pts) Find an algebraic expression in x that is equivalent to sec(arcsin( 2x )). a. q 1 1 2 Faresin E 0 4x q 1 2 Then sin o II b. 1+ 4x 0 in EE p 1 2 c. 1+4x2 p d. p 2 2x2 1 FE I fill in using e. 4 x2 pythagorean theorem 9. (2 pts) Find all solutions in radians of sin2 x = sin x in the interval [0, 2p ). a. 0, p 2, 2p sin x sin x 0 b. 0, p 2, 3p c. 0, p 3, p Sin x sin x 1 0 d. 0, p 4, p e. 0, p 2, p Sin x 0 1 0 si x 0 I sin x I I 10. (2 pts) Find the exact value of sin( p3 + p4 ). p p 2(1 3) a. 4 sin sin E cos E cos E sin p p 2( 1 3) b. 4 c. p p 2( 3 1) 4 E E E E p p 2(1+ 3) d. 4 p p 3(1 2) e. 4 11. (2 pts) Use the figure below to find the exact value of csc(2q ). A 62T 37 1 q B 6 C Line 37 a. 12 first 37 b. 35 c. 12 Sin20 2sinocos0 37 d. 35 37 É 12 e. 35 fliptoget csc 20 12. (2 pts) Which of the following expressions is equivalent to 1 2 sin2 x + sin4 x? a. sin2 x b. cos2 x sin x 2sink I sin x 1 sin x 1 c. sin4 x sin x 1 d. cos4 x e. sin4 x + cos2 x cos_x Cos x 13. (10 pts) Solve 2 sin2 x 1 = 0 for all x in [0, 2p ). Your answers should be in radians! Solution: I 25in x 4 3 E E 25 I 5in x ñ I II If I E E sin x I F IT 312 14. (10 pts) Solve csc q = 2 for all x in [0, 4p ). Your answers should be in radians! note the interval Solution: silo 21 The 216 0 715 19 a 2 0 716 1961 I 15. (10pts) Use identities to solve exactly the trigonometric equation p 3 cos( x ) cos(2x ) + sin( x ) sin(2x ) = 2 over the interval [0, 2p ). Your answers should be in radians!. Solution: cos x cos 2x sin x sin 2x 10s x 2x 52 COS x 25 COS x cos x I 2 I 2 16. (15 pts) Consider the Pythagorean Theorem x2 + y2 = r2. (a) (7 pts) Draw a right triangle, labeling x, y, r and q clearly in your picture. Recall that the side adjacent to q is taken to be x by convention. Solution: r y O r (b) (8 pts) Using your picture and the Pythagorean Theorem, find an equation that relates sec q and tan q. Explain how you know your resulting equation is true. Twyourworker Solution: my c 2 I E I tanto sec lo aÉgsecco and tan O 17. (15 points) Verify that the equation below is in fact an identity: sec2 ( x ) 1 2 = sin2 ( x ) sec ( x ) mm Solution: sec x 1 tank sec x co's x cosex I iÉ 18. (15 pts) In the triangle shown in the figure below, find the measure of \ A, where AB = 8, BC = 7, and AC = 5. Note: The figure is not drawn to scale. C s 7 A B 8 Solution: Aca Apr 2 AC AB COS A 72 52 82 2 5 8 Cos A 49 25 64 80COS A 49 89 80COS A 40 80COS A f.fi w̅ COS A angle must be between 0 and 180 or Oand i since it is in a Δ Solution: Extra Page Fundamental Identities cos2 (q ) + sin2 (q ) = 1, tan2 (q ) + 1 = sec2 (q ), 1 + cot2 (q ) = csc2 (q ) Addition Formulas sin(a + b) = sin(a) cos( b) + cos(a) sin( b) cos(a + b) = cos(a) cos( b) sin(a) sin( b) Subtraction Formulas sin(a b) = sin(a) cos( b) cos(a) sin( b) cos(a b) = cos(a) cos( b) + sin(a) sin( b) Double Angle Formulas sin(2a) = 2 sin(a) cos(a) cos(2a) = cos2 (a) sin2 (a) = 1 2 sin2 (a) = 2 cos2 (a) 1 Laws of Sines and Cosines sin A sin B sin C = = a b c c2 = a2 + b2 2ab cos C