Mth 133 Trigonometry Review Problems for the Final Examination

Transcription

1 Mth 1 Trigonometry Review Problems for the Final Examination Thomas W. Judson Stephen F. Austin State University Fall 017 Final Exam Details The final exam for MTH 1 will is comprehensive and will cover the entire course. Calculators will be allowed on this exam. To find out when your final exam will, consult the final exam schedule at If you qualify for an accommodation, please make arrangements with Disability Services as soon as possible. If you wait until the day of the exam to make arrangements, we will not be able to meet your accommodation needs. Studying and Reviewing You should try working some of the problems in the review sections at the end of each chapter. Be sure to take advantage of the office hours, SI sessions, and the AARC. 1

2 Sample Exam Questions The following exam questions are from Exams 1,, and. These questions are not meant to be inclusive and there will be questions on the final exam from the material covered since Exam. 1. Indicate whether each of the following statements are true or false. a The angles / and 8/ are coterminal. b The angle 770 terminates in quadrant IV quadrant. c The degree measure of an angle cannot be negative. d A central angle of one radian in a unit circle intercepts an arc of length one. e The length of the arc intercepted by a central angle of / radians on a circle of radius is. f If sinθ < 0 and tanθ < 0, then θ is in quadrant III quadrant. g The reference angle for 7/ is /. h As the earth rotates, Miami and Boston have the same angular velocity. i As the earth rotates, Miami and Boston have the same linear velocity. j The phase shift of y = sinx + / is /. Page

3 a ; b ; c ; d ; e ; f ; g ; h ; i ; j.. a Convert 7/10 radians to degrees. Be sure to simplify and show your work = 1 b Name the quadrants in which the tangent function has negative values. Quadrants II and IV c What is the reference angle for α = / radians? Leave your answer in terms of.. If sin θ = 1/ and θ is an angle in Quadrant II, find the exact values of the other five trig functions of θ. cos θ = tan θ = cot θ = sec θ = csc θ = cos θ = 15/ tan θ = 1/ 15 cot θ = / 15 sec θ = 15 csc θ =. Draw an angle θ in standard position on the unit circle whose terminal side contains the point /5, /5. Then, find exact values of the six trigonometric functions of θ. Page

4 y x sin θ = cos θ = tan θ = cot θ = sec θ = csc θ = y /5, /5 x sin θ = /5 cos θ = /5 tan θ = / cot θ = / sec θ = 5/ csc θ = 5/ 5. The London Eye shown below can carry 800 passengers in capsules around a circle measuring meters in circumference. If one revolution takes 0 minutes, then what is the linear velocity of a capsule in feet per second rounded to the nearest tenth? Page

5 0.8 feet per second. On the figure below, write the coordinates of the points on the unit circle which lie on the terminal sides of the angles shown. y 0, , ,0 x ,-1 Page 5

6 y,, 1 1, , ,1 90 1, 0 5 0,, 1 0 1,0 x , 1 0, 1, 70 0, , 1 1,, Use the points on the unit circle above to determine the exact value for each of the following. If the expression is undefined, then state such. a sec = c sin 5 = 7 5 b csc = d tan = a sec is undefined. b csc 7 = c sin 5 = 5 d tan = 1 8. A 100-ft guy wire is attached to the top of a tall antenna. The angle between the guy wire and the ground is. How tall is the antenna to the nearest foot. Page

7 88 ft 9. A slice of pizza with a central angle of /7 is cut from a pizza of radius 10 in. What is the area of the slice to the nearest tenth of an inch?.5 square inches. 10. Indicate whether each of the following statements are true or false. a cot = 1/ tan. b tan x = sec x csc x. c sin 9 + cos 9 = 1. d The domain of tan 1 x is,. e If x is in [ 1, 1], then sin 1 x is in [, ]. f sin 1 x = sinx 1. a ; b ; c ; d ; e ; f ; 11. Give the period, phase shift and range for each function given below. Page 7

8 function period phase shift range y = cos x 7 1 y = sin x y = tan x y = sec x function period phase shift range y = cos x [, 1] y = sin x 8 [, ] y = tan x 0, y = sec x + + 1, 0] [, 1. Sketch the graph of one complete cycle of y = sin x +. Clearly indicate the coordinates of the key points in the cycle. The five key points are the starting point, maximum point, middle point, minimum point and ending point. x starting point maximum point middle point minimum point ending point y Page 8

9 x y Find an equation of a sine or cosine function which is represented by the curve below. 1, , 1,, 1 Page 9

10 y = sin x 1 1. Find the exact value of the composition tan cos 1 /. Do not give calculator approximations Use identities to rewrite the following expressions as a single function of a single angle. Do not calculator approximations. a sin 7 cos 10 + cos 7 sin 10 = b sin 11 cos 9 cos 11 sin 9 = c cos sin = 1 1 d sin cos = e tan 1 tan tan 1 tan 8 = a sin 7 cos 10 + cos 7 sin 10 = sin17 b sin 11 cos 9 cos 11 sin 9 = sin c cos sin = cos/7 1 1 d sin cos = sin 5 e tan 1 tan tan 1 tan 8 = tan Page 10

11 1. Show your steps and justify each step as you prove that the following equation is an identity. csc x cos x cot x = sin x csc x cos x cot x = 1 cos x cos x sin x sin x = 1 cos x cos x = sin x sin x = sin x 17. If sin θ = /5 and θ is an angle in Quadrant II, find the exact value of sinθ. Simplify your answer. Do not give calculator approximations Suppose that α and β are angles with the following sine and cosine values. cos α = 8 sin α = 1 cos β = 1 1 sin β = 5 1 Use appropriate identities to find the exact values of cosα β. Simplify your answer. Do not give calculator approximations = Indicate whether each of the following statements are true or false. a If x = + k for any integer k, then x = + k. b The solution set to tan x = 1 is {x x = 7 + k}. c The only solution to sinα = 0.55 in the interval [0, is sin Page 11

12 d If you know the measures of all three angles of a triangle, then you can use the law of sines to find the lengths of the sides. e The smallest angle of a triangle lies opposite the shortest side. f If the largest angle of a triangle is obtuse, then the other two are acute. g The magnitude of vector v + w is the sum of the magnitudes of vectors v and w. h The magnitude of vector v is equal to the magnitude of vector v. i The area of a triangle can be found if you know the lengths of the three sides. j The area of a triangle is one-half the product of the lengths of any two sides. a ; b ; c ; d ; e ; f ; g ; h ; i ; j. 0. Find all real solutions x that satisfy the following equations. Use k to represent any integer. Give exact values in radian measure. a sin x + =. x = + k 5 + k b cosx + 1 = 0. x = + k + k Page 1

13 1. Find all real solutions x that satisfy the equation sin x sin x 1 = 0 in the interval [0, ]. Give exact values in radian measure. x = 5 7. Find all possible values for β in a triangle for which side a = 10, side b = 1 and angle α = 0. Round to the nearest tenth of a degree. β = Find the side length a in the triangle that has side b =, side c = 19 and angle α = 7. Round to the nearest tenth of a unit. a = Find the measure of the largest angle in a triangle with sides a = 8, b = and c = 7. Round to the nearest tenth of a degree. largest angle = Find the component form v = a, b of a vector with magnitude equal to 8 and direction angle equal to 150. Give exact values. v = Page 1

14 ,. For the vectors v =, 5 and w =,, perform the indicated operations. a v b v + w c v w d v w e Compute the angle θ between vectors v and w Round to the nearest tenth of a degree θ = a 9 b 5, 1 c 1 d 1, 1 e Sketch the requested vectors using the given drawings. a Sketch u + v b Sketch w u w v Page 1