2018 Midterm Review Trigonometry: Midterm Review A Missive from the Math Department Trigonometry Work Problems Study For Understanding Read Actively

Transcription

1 Summer

2 . Fill in the blank to correctl complete the sentence..4 written in degrees and minutes is..4 written in degrees and minutes is.. Find the complement and the supplement of the given angle. The complement of the angle is. The supplement of the angle is.. Perform the calculation. 48 7' 4' 48 7' 4 ' = ' (Simplif our answers. Tpe an integer or a fraction.) 4. Convert the angle to degrees, minutes, and seconds notation =. The terminal side of an angle θ in standard position passes through the point (, ). Use the figure to find the following value. sin θ - θ - sin θ = (Tpe an eact answer in simplified form. Rationalize all denominators.)

3 . Sketch an angle θ in standard position such that θ has the least possible positive measure, and the point, is on the terminal side of θ. Then find the values of the si trigonometric functions for the angle. Rationalize denominators if applicable. Do not use a calculator. Choose the correct graph below. B. C. D Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. sin θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) B. The function is undefined. Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. cos θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) B. The function is undefined. Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. tan θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) B. The function is undefined. Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. sec θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) B. The function is undefined. Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. csc θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) B. The function is undefined. Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. cot θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.)

4 B. The function is undefined. 7. The equation, with a restriction on, is the terminal side of an angle θ in standard position. + = 0, 0 Give the eact values of the si trigonometric functions of θ. Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. sin θ = (Simplif our answer. Use integers or fractions for an numbers in the epression. Tpe an eact answer, using radicals as needed. Rationalize all denominators.) B. The function is undefined. Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. cos θ = (Simplif our answer. Use integers or fractions for an numbers in the epression. Tpe an eact answer, using radicals as needed. Rationalize all denominators.) B. The function is undefined. Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. tan θ = (Simplif our answer. Use integers or fractions for an numbers in the epression. Tpe an eact answer, using radicals as needed. Rationalize all denominators.) B. The function is undefined. Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. csc θ = (Simplif our answer. Use integers or fractions for an numbers in the epression. Tpe an eact answer, using radicals as needed. Rationalize all denominators.) B. The function is undefined. Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. sec θ = (Simplif our answer. Use integers or fractions for an numbers in the epression. Tpe an eact answer, using radicals as needed. Rationalize all denominators.) B. The function is undefined. Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. cot θ = (Simplif our answer. Use integers or fractions for an numbers in the epression. Tpe an eact answer, using radicals as needed. Rationalize all denominators.) B. The function is undefined.

5 8. Find the indicated function value. If it is undefined, sa so. csc 00 Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. csc 00 = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) B. The function is undefined. 9. Evaluate the trigonometric function at the quadrantal angle, or state that the epression is undefined. cos ( 90 ) Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. cos ( 90 ) = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) B. The function is undefined. 0. Select the correct choices that complete the sentence below. Given tan θ =, two equivalent forms of this identit are cot θ = () and cot θ tan θ () =. () tan θ () cot θ tan θ tan θ cot θ tan θ cot θ cot θ. Select the correct choices that complete the sentence below. For an angle θ measuring 0, the trigonometric functions () and () are positive and the remaining trigonometric functions are negative. () tan θ () sin θ cos θ sec θ csc θ cot θ 4

6 . Decide whether the statement is possible or impossible. cos θ < 0 and sec θ < 0 Is the statement possible or impossible? The statement is possible because sec θ is alwas negative. B. The statement is impossible because cosine and secant alwas have opposite signs. C. The statement is possible because cos θ is alwas negative. D. The statement is impossible because sec θ is alwas positive. E. The statement is possible because cosine and secant alwas have the same sign. F. The statement is impossible because cos θ is alwas positive.. Find the signs of the si trigonometric function values for the given angle. 7 sin 7 is () cos 7 is () and tan 7 is () csc 7 is (4) sec 7 is () and cot 7 is () () negative, () positive, () negative. (4) negative, () positive, positive, negative, positive. positive, negative, () positive. negative. 4. Identif all possible quadrants of an angle θ that satisfies the given conditions. tan θ < 0, sin θ > 0 Select all possible quadrants below. Quadrant III B. Quadrant IV C. Quadrant II D. Quadrant I. Find the eact value of csc θ, given that cot θ = and θ is in quadrant IV. Rationalize denominators when applicable. Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. csc θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) B. The function is undefined.

7 . Find the eact value of tan θ, given that sin θ = and θ is in quadrant I. Rationalize denominators when applicable. Select the correct choice below and, if necessar, fill in the answer bo to complete our choice. tan θ = (Simplif our answer. Tpe an eact answer, using radicals as needed. Tpe an integer or a fraction.) B. The function is undefined. 7. Suppose ABC is a right triangle with sides a, b, and c and right angle at C. Use the Pthagorean theorem to find the unknown side length. Then find the values of the si trigonometric functions for angle B. Rationalize the denominators when applicable. c B a a =, c = A b C The unknown side length b is. (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) sin B = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) cos B = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) tan B = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) sec B = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) csc B = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) cot B = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) 8. Rewrite cos 80 in terms of its cofunction. cos 80 = (Tpe an eact answer. Simplif our answer. Tpe an angle measures in degrees. Do not include the degree smbol in our answer.) 9. Find a solution for the equation. Assume that all angles in which an unknown appears are acute angles. cot α = tan( α + 0 ) α = (Simplif our answer. Tpe an integer or a decimal.) 0. Find the eact value of tan 0. tan 0 = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.)

8 . Complete the table with eact trigonometric function values. Do not use a calculator. θ sin θ cos θ tan θ cot θ sec θ csc θ 40??? θ sin θ cos θ tan θ cot θ sec θ csc θ 40 (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.). Evaluate the epression. sin 0 cos 70 + tan 4 sin 0 cos 70 + tan 4 = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.). Find all values of θ in the interval [0,0 ) that have the given function value. sin θ = θ = (Tpe an integer or a decimal. Use a comma to separate answers as needed.) 4. Find all values of θ in the interval [0,0 ) that have the given function value. tan θ = θ = (Tpe an integer or a decimal. Use a comma to separate answers as needed.). Find the coordinates of the point P on the circumference of the circle. (Hint: Add - and -aes, assuming that the angle is in standard position.) P 0 0 The coordinates of P are. (Tpe an ordered pair. Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) 7

9 . Match the trigonometric function with its appropriate approimation. sec 7 Choose the correct answer below. sec B. sec C. sec D. sec Find a value of θ in the interval [0,90 ] that satisfies the given statement. sin θ = θ (Simplif our answer. Tpe an integer or a decimal. Round to si decimal places if needed.) 8. Match the following equation with the appropriate right triangle. The goal is to find the value of. = 7 sin 40 Choose the correct triangle below. 7 7 B. C D. 7 E. F A surveor measured the road frontage of a parcel of land and obtained a distance of. ft. Based on this result, the distance must be between what two values? The distance must be at least The distance must be less than ft. ft. 0. Solve the right triangle. B a 9 m C b 9 9' A B = ' (Round to the nearest integer as needed.) a = m (Round to the nearest integer as needed.) b = m (Round to the nearest integer as needed.) 8

10 . Solve the right triangle ABC, where C = 90. Give angles in degrees and minutes. a = 8. cm, c = 4. cm b cm (Round to the nearest tenth as needed.) A = (Round to the nearest minute as needed.) B = (Round to the nearest minute as needed.). A.-m fire truck ladder is leaning against a wall. Find the distance d the ladder goes up the wall (above the fire truck) if the ladder makes an angle of 44 7' with the horizontal. d m (Simplif our answer. Tpe an integer or a decimal. Round to the nearest hundredth.). m d 44 7'. The shadow of a vertical tower is 7.0 ft long when the angle of elevation of the sun is.0. Find the height of the tower. The tower is ft tall. (Simplif our answer. Tpe an integer or decimal rounded to the nearest tenth as needed.) 4. A compan safet committee has recommended that a floodlight be mounted in a parking lot so as to illuminate the emploee eit. Find the angle of depression of the light to the nearest minute. angle of depression θ 9.ft θ angle of elevation.4ft The angle of depression of the light to the nearest minute is.. Match the measure of bearing with the appropriate graph. Choose the correct answer below. B. C. D. N N N N W E W E W E W E S S S S 9

11 . An observer for a radar station is located at the origin of a coordinate sstem. Find the bearing of an airplane located at the point (, ). Epress the bearing using both methods One epression for the bearing uses a single angle measure. The bearing using this method is. Another epression for the bearing gives the direction from a north-south line. What is the bearing using this method? N 4 W B. S 4 E C. N 4 E D. S 4 W A ship leaves its port and sails on a bearing of N 0'E, at speed 8. mph. Another ship leaves the same port at the same time and sails on a bearing of S8 40'E, at speed.0 mph. 0' c 8 40' How far apart are the ships after hours? miles (Round to the nearest integer as needed.) Convert the following degree measure to radian measure. 0 Convert the degree measure below to radians. 0 0 = radians (Simplif our answer. Tpe an eact answer, using as needed. Use integers or fractions for an numbers in the epression.) 0 = radians (Simplif our answer. Tpe an eact answer, using epression.) as needed. Use integers or fractions for an numbers in the Convert the following radian measure to degree measure. 4 Convert the following radian measure to degrees = radians = (Round to the nearest minute as needed.) 0

12 4. Find the eact value of the following epression without using a calculator. sin sin = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) 4. Find the eact value of the following epression without using a calculator. sec 9 sec 9 = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) 44. Through how man radians will the hour hand on a clock rotate in (a) hr? In (b) hr? (a) In hr, the hour hand on the clock will rotate radians. (Tpe an eact answer in terms of. Tpe an integer or a simplified fraction.) (b) In hr, the hour hand on the clock will rotate radians. (Tpe an eact answer in terms of. Tpe an integer or a simplified fraction.) 4. A stationar horse on a carousel makes complete The horse revolves through radians revolutions. Through what radian measure does the horse (Simplif our answer. Use integers or fractions for an revolve? numbers in the epression. Tpe an eact answer, using as needed.) 4. Find the radius of the circle in the figure to the right. The radius of the circle is. (Tpe an integer or a simplified fraction.) 47. Find the measure (in radians) of the central angle. The area of a sector is sq units. 4 Picture is not drawn to scale The measure is radian(s). (Tpe an integer or decimal rounded to the nearest tenth as needed.)

13 48. Find the length to three significant digits of the arc intercepted b a central angle θ in a circle of radius r. r =.9 in., θ = 77 The length of the intercepted arc is approimatel in. 49. The figure shows the chain drive of a biccle. How far will the biccle move if the pedals are rotated through 80? Assume the radius of the biccle wheel is. inches..8 in 4.47 in The biccle will travel approimatel (Round to the nearest tenth.) in. 0. Find the area of the sector of a circle having radius r and central angle θ. r =. cm, θ = The area is approimatel cm. (Round to the nearest tenth.). Find the measure (in radians) of a central angle of a sector of area 8 square inches in a circle of radius.9 inches. The central angle measures approimatel (Round to the nearest tenth.) radians.. A frequent problem in surveing cit lots and rural lands adjacent to curves of highwas and railwas is that of finding the area when one or more of the boundar lines is the arc of a circle. Find the area of the lot shown in the figure d 0 d The area is d. (Round to the nearest whole number.)

14 . The figure shows an angle θ in standard position with its terminal side intersecting the unit circle. Evaluate the si circular function values of θ. θ, sin θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) cos θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) tan θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) sec θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) csc θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) cot θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.)

15 4. Use the unit circle, along with the definitions of the circular functions, to find the eact values for the given functions sin, cos, tan when s =. Select the correct choice below and fill in an answer boes in our choice. sin = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression. ) B. The solution is undefined. Select the correct choice below and fill in an answer boes in our choice. cos = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression. ) B. The solution is undefined. What is the value of tan? 0 B. C. The solution is undefined D.. Find the eact value of csc. csc = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.). Find the eact function value. tan 4 tan 4 = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) 7. Find a value of s in the interval given statement. 0, that satisfies the s = (Round to eight decimal places as needed.) sec s =.7 4

16 8. In the figure, point P moves at a constant rate along the unit circle. Complete the sentence below. If the angular speed of point P is radian per second, then P will move around the entire circle in sec. P θ O r B If the angular speed of point P is radian per second, then P will move around the entire circle in sec. (Simplif our answer. Tpe an eact answer, using epression.) as needed. Use integers or fractions for an numbers in the 9. Suppose that point P is on a circle with radius r, and ra OP is rotating with angular speed ω. Complete parts (a) through (c) for the given values of r, ω, and t. r = in., ω = radian per min, t = min (a) What is the angle generated b P in time t? θ = radians (Simplif our answer. Tpe an eact answer, using epression.) (b) What is the distance traveled b P along the circle in time t? s = inches (Simplif our answer. Tpe an eact answer, using epression.) (c) What is the linear speed of P? v = inches per minute (Simplif our answer. Tpe an eact answer, using epression.) as needed. Use integers or fractions for an numbers in the as needed. Use integers or fractions for an numbers in the as needed. Use integers or fractions for an numbers in the 0. Use the formula ω = θ t to find the value of the missing variable. θ = 7 radians, ω = radian per minute t = min (Simplif our answer. Tpe an integer or a fraction.). The formula ω = θ can be rewritten as θ = ωt. Using ωt for θ changes s = rθ to s = rωt. Use the formula s = rωt to find the t value of the missing variable. s = cm, r = cm, ω = radian per sec t = sec (Tpe an integer or a fraction.)

17 . Find the velocit, v, of the tip of the minute hand of a clock, if the hand is 0 cm long. v = cm per minute (Simplif our answer. Tpe an eact answer, using equation.) as needed. Use integers or fractions for an numbers in the. The tires of a biccle have radius 0.0 in. and are turning at the rate of 0 revolutions per min. See the figure. How fast is the biccle traveling in miles per hour? (Hint: 80 ft = mi) 0.0 in. How fast is the biccle traveling? mph (Tpe an integer or decimal rounded to the nearest tenth as needed.) 4. The pulle shown has a radius of.8 cm. Suppose it takes sec for 7 cm of belt to go around the pulle. (a) Find the linear speed of the belt in centimeters per second. (b) Find the angular speed of the pulle in radians per second..8 cm (a) The linear speed of the roller is cm per second. (Tpe an integer or decimal rounded to the nearest tenth as needed.) (b) The angular speed of the roller is radian(s) per second. (Tpe an integer or decimal rounded to the nearest hundredth as needed. Use the answer from part (a) to find this answer.). Match the function = sin with its graph. Which is the correct graph of the equation? B. C

18 . Determine the amplitude of the function = (). Also, choose its graph. cos The amplitude is. Which is the correct graph of the equation on the interval [, ]? B. C. D Graph the following function over a two-period interval. Give the period and the amplitude. = sin What is the period of the function = sin? (Simplif our answer. Tpe an eact answer, using epression.) What is the amplitude of the function = sin? (Tpe an integer or a simplified fraction.) Which of the following is the correct graph of the function = sin? as needed. Use integers or fractions for an numbers in the B. C. D

19 8. Graph the following function over a two-period interval. Give the period and the amplitude. = cos 4 What is the period of the function = cos 4? (Simplif our answer. Tpe an eact answer, using epression.) What is the amplitude of the function = cos 4? (Tpe an integer or a simplified fraction.) Which of the following is the correct graph of the function = cos 4? as needed. Use integers or fractions for an numbers in the B. C. D The function graphed to the right is of the form = a sin b or = a cos b, where b > 0. Determine the equation of the graph. = (Tpe an epression using as the variable Find the graph of one period of the function below. = cos Choose the graph of = cos. B. C. D

20 7. Determine the equation of the function that is graphed. Choose the correct answer below. 0 8 = cos + B. = cos C. = cos + D. = cos The function graphed to the right is of the form = cos ( d) or = sin ( d), where d is the least positive value. Determine the equation of the graph Choose the correct equation of the graph. = cos + B. = sin + C. = cos D. = sin 7. Determine the amplitude, period, an vertical translation, and an phase shift of the given graph. = sin The amplitude is. The period is. (Tpe an eact answer, using as needed.) Select the correct choice below and fill in an answer boes within our choice. The vertical translation is units. B. There is no vertical translation. Select the correct choice below and fill in an answer boes within our choice. The phase shift is units to the left. (Tpe an eact answer, using B. There is no phase shift. as needed.) 9

21 74. Graph the following function. = sin Use the graphing tool to graph the function. (For an answer boes shown with the grapher, tpe an eact answer. Tpe the word pi to insert the smbol as needed.) - 7. Graph the function over a one-period interval. = tan Choose the correct graph. B. C. D Graph the function over a one-period interval. = tan Which graph below shows one period of the function? B. C. D.,, 4, 4, ,,, 4 8, 0

22 77. Graph the function over a two-period interval. = cot ( + ) Which graph below shows two periods of the function? B. C. D Find the graph of = 4 csc. Choose the correct graph. B. C. D Graph the function. = csc + 4 Choose the correct graph below. B. C. D The function graphed to the right is of the form = a sec b + c or = a csc b + c for some a 0, b > 0. Determine the equation of the function An equation of the function shown is =.

23 sin θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) cos θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) tan θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) sec θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) csc θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) cot θ = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.)

24 7. sin θ = (Simplif our answer. Use integers or fractions for an numbers in the epression. Tpe an eact answer, using radicals as needed. Rationalize all denominators.) cos θ = (Simplif our answer. Use integers or fractions for an numbers in the epression. Tpe an eact answer, using radicals as needed. Rationalize all denominators.) tan θ = (Simplif our answer. Use integers or fractions for an numbers in the epression. Tpe an eact answer, using radicals as needed. Rationalize all denominators.) csc θ = (Simplif our answer. Use integers or fractions for an numbers in the epression. Tpe an eact answer, using radicals as needed. Rationalize all denominators.) sec θ = (Simplif our answer. Use integers or fractions for an numbers in the epression. Tpe an eact answer, using radicals as needed. Rationalize all denominators.) cot θ = (Simplif our answer. Use integers or fractions for an numbers in the epression. Tpe an eact answer, using radicals as needed. Rationalize all denominators.) 8. B. The function is undefined. 9. cos ( 90 ) = 0 (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression.) 0. () tan θ () cot θ. () sin θ () csc θ. E. The statement is possible because cosine and secant alwas have the same sign.

25 . () positive, () negative, () negative. (4) positive, () negative, () negative. 4. C. Quadrant II. csc θ = epression.) (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the. tan θ = (Simplif our answer. Tpe an eact answer, using radicals as needed. Tpe an integer or a fraction.) sin

26 ,0 4.,.,. D. sec D

27 N W E B. S. D. S 4 W

28 sin = (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression. ) cos = 0 (Simplif our answer, including an radicals. Use integers or fractions for an numbers in the epression. ) C. The solution is undefined. 7

29 B B. - 8

30 7. D D sin 70. C B. = cos 7. C. = cos 7. 8 B. There is no vertical translation. The phase shift is units to the left.(tpe an eact answer, using as needed.) 9

31 , C., C

32 79. D sec ()