Skip to main content

Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)

13RE 14RE Angular Speed and Linear Speed A potters wheel with radius 8 in. spins at 150 rpm. Find the angular and linear speeds of a point on the rim of the wheel.16RE 17RE 18RE 19RE 20RE 21RE Finding Sides in Right Triangles Find the sides labeled x and y, rounded to two decimal places. 22.Solving a Triangle Solve the triangle. 23.Solving a Triangle Solve the triangle. 24.25RE 26RE Trigonometric Ratios Express the lengths a and b in the figure in terms of the trigonometric ratios of .28RE 29RE Pistons of an Engine The pistons in a car engine move up and down repeatedly to turn the crankshaft, as shown. Find the height of the point P above the center O of the crankshaft in terms of the angle .Radius of the Moon As viewed from the earth, the angle subtended by the full moon is 0.518. Use this information and the fact that the distance AB from the earth to the moon is 236,900 mi to find the radius of the moon.Distance Between Two Ships A pilot measures the angles of depression to two ships to be 40 and 52 (see the figure). If the pilot is flying at an elevation of 33,000 ft, find the distance between the two ships.Values of Trigonometric Functions Find the exact value. 33. sin 315 34RE 35RE 36RE 37RE 38RE 39RE 40RE 41RE 42RE 43RE 44RE 45RE 46RE 47RE 48RE 49RE 50RE Expressing One Trigonometric Function in Terms of Another Write the first expression in terms of the second, for in the given quadrant. 51. tan2 , sin ; in any quadrant 52RE 53RE 54RE 55RE 56RE 57RE 58RE 59RE 60RE 61RE 62RE 63RE 64RE 65RE 66RE 67RE 68RE 69RE 70RE 71RE 72RE 73RE 74RE 75RE 76RE 77RE 78RE Distance Between Two Ships Two ships leave a port at the same time. One travels at 20 mi/h in a direction N 32 E, and the other travels at 28 mi/h in a direction S 42 E (see the figure). How far apart are the two ships after 2 h?Height of a Building From a point A on the ground, the angle of elevation to the top of a tall building is 24.1. From a point B, which is 600 ft closer to the building, the angle of elevation is measured to be 30.2. Find the height of the building.Distance Between Two Points Find the distance between points A and B on opposite sides of a lake from the information shown.Distance Between a Boat and the Shore A boat is cruising the ocean off a straight shoreline. Points A and B are 120 mi apart on the shore, as shown. It is found that A=42.3 and B=68.9. Find the shortest distance from the boat to the shore.83RE 84RE Find the radian measures that correspond to the degree measures 330 and 135.2T 3T Find the exact value of each of the following. (a) sin 405 (b) tan(150) (c) sec53 (d) csc52 Find tan + sin for the angle shown.Express the lengths a and b shown in the figure in terms of .If cos=13 and is in Quadrant III, find tan cot + csc .8T Express tan in terms of sec for in Quadrant II.The base of the ladder in the figure is 6 ft from the building, and the angle formed by the ladder and the ground is 73. How high up the building does the ladder touch?Express in each figure in terms of x. (a) (b)Find the exact value of cos(tan1910).13T Find the side labeled x or the angle labeled . 14.Find the side labeled x or the angle labeled . 15.16T 17T 18T Refer to the figure below. (a) Find the area of the shaded region. (b) Find the perimeter of the shaded region.Refer to the figure below. (a) Find the angle opposite the longest side. (b) Find the area of the triangle.Two wires tether a balloon to the ground, as shown. How high is the balloon above the ground?1P 2P Determining a Distance A surveyor on one side of a river wishes to find the distance between points A and B on the opposite side of the river. On her side she chooses points C and D, which are 20 m apart, and measures the angles shown in the figure below. Find the distance between A and B.Height of a Cliff To measure the height of an inaccessible cliff on the opposite side of a river, a surveyor makes the measurements shown in the figure at the left. Find the height of the cliff.Height of a Mountain To calculate the height h of a mountain, angles and and distance d are measured, as shown in the figure below. (a) Show that h=dcotcot (b) Show that h=dsinsinsin() (c) Use the formulas from parts (a) and (b) to find the height of a mountain if = 25, = 29, and d = 800 ft. Do you get the same answer from each formula?Determining a Distance A surveyor has determined that a mountain is 2430 ft high. From the top of the mountain he measures the angles of depression to two landmarks at the base of the mountain and finds them to be 42 and 39. (Observe that these are the same as the angles of elevation from the landmarks as shown in the figure at the left.) The angle between the lines of sight to the landmarks is 68. Calculate the distance between the two landmarks.Surveying Building Lots A surveyor surveys two adjacent lots and makes the following rough sketch showing his measurements. Calculate all the distances shown in the figure, and use your result to draw an accurate map of the two lots.8P An equation is called an identity if it is valid for __________ values of the variable. The equation 2x = x + x is an algebraic identity, and the equation sin2 x + cos2 x = __________ is a trigonometric identity.For any x it is true that cos(x) has the same value as cos x. We express this fact as the identity _______________.Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. 3. cos t tan t Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. 4. cos t csc t 5E 6E Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. 7. tan2 x sec2 x 8E Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. 9. sin u + cot u cos u 10E Simplifying Trigonometric Expressions Write the trigonometric expression in terms of sine and cosine, and then simplify. 11. seccossin 12E 13E 14E 15E Simplifying Trigonometric Expressions Simplify the trigonometric expression. 16. 1+cotAcscA 17E Simplifying Trigonometric Expressions Simplify the trigonometric expression. 18. sin4 cos4 + cos2 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E Proving an Identity Algebraically and Graphically Consider the given equation. (a) Verify algebraically that the equation is an identity. (b) Confirm graphically that the equation is an identity. 29. cosxsecxsinx=cscxsinx 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E Proving Identities Verify the identity. 41. (1cos)(1+cos)=1csc2 42E 43E 44E 45E 46E 47E 48E Proving Identities Verify the identity. 49. csc x cos2 x + sin x = csc x Proving Identities Verify the identity. 50. cot2 t cos2 t = cot2 t cos2 t Proving Identities Verify the identity. 51. (sinx+cosx)2sin2xcos2x=sin2xcos2x(sinxcosx)2 Proving Identities Verify the identity. 52. (sin x + cos x)4 = (1 + 2 sin x cos x)2 53E 54E 55E 56E 57E 58E 59E 60E 61E 62E Proving Identities Verify the identity. 63. secx+cscxtanx+cotx=sinx+cosx 64E 65E 66E Proving Identities Verify the identity. 67. tan2 u sin2 u = tan2 u sin2 u Proving Identities Verify the identity. 68. sec4 x tan4 x = sec2 x + tan2 x 69E 70E 71E 72E 73E 74E 75E 76E 77E 78E 79E 80E 81E 82E Proving Identities Verify the identity. 83. cos1sin=sec+tan 84E 85E 86E 87E 88E Trigonometric Substitution Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7). Assume that 0 /2. 89. x1x2, x = sin Trigonometric Substitution Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7). Assume that 0 /2. 90. 1+x2, x = tan Trigonometric Substitution Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7). Assume that 0 /2. 91. x21, x = sec Trigonometric Substitution Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7). Assume that 0 /2. 92. 1x24+x2, x = 2 tan 93E 94E 95E Determining Identities Graphically Graph f and g in the same viewing rectangle. Do the graphs suggest that the equation f(x) = g(x) is an identity? Prove your answer. 96. f(x) = tan x(1 + sin x), g(x)=sinxcosx1+sinx Determining Identities Graphically Graph f and g in the same viewing rectangle. Do the graphs suggest that the equation f(x) = g(x) is an identity? Prove your answer. 97. f(x) = (sin x + cos x)2, g(x) = 1 98E 99E 100E 101E 102E 103E 104E 105E 106E 107E 108E 109E 110E 111E 112E 113E DISCUSS: Equations That Are Identities You have encountered many identities in this course. Which of the following equations do you recognize as identities? For those that you think are identities, test several values of the variables to confirm that the equation is true for those variables. (a) (x + y)2 = x2 + 2xy + y2 (b) x2 + y2 = 1 (c) x(y + z) = xy + xz (d) t2 cos2 t = (t cos t)(t + cos t) (e) sin t + cos t = 1 (f) x2 tan2 x = 0 115E 116E 117E DISCUSS: Cofunction Identities In the right triangle shown, explain why v = (/2) u. Explain how you can obtain all six cofunction identities from this triangle for 0 u /2. Note that u and v are complementary angles. So the cofunction identities state that a trigonometric function of an angle u is equal to the corresponding cofunction of the complementary angle v.If we know the values of the sine and cosine of x and y, we can find the value of sin(x + y) by using the _____ Formula for Sine. State the formula: sin(x + y) = __________.If we know the values of the sine and cosine of x and y, we can find the value of cos(x y) by using the _____ Formula for Cosine. State the formula: cos(x y) = __________.3E 4E 5E Values of Trigonometric Functions Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. 6. cos 195 7E Values of Trigonometric Functions Use an Addition or Subtraction Formula to find the exact value of the expression, as demonstrated in Example 1. 8. tan 165 9E 10E 11E 12E 13E 14E Values of Trigonometric Functions Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value. 15. sin 18 cos 27 + cos 18 sin 27 Values of Trigonometric Functions Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value. 16. cos 10 cos 80 sin 10 sin 80 Values of Trigonometric Functions Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value. 17. cos37cos221+sin37sin221 Values of Trigonometric Functions Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value. 18. tan18+tan91tan18tan9 19E Values of Trigonometric Functions Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value. 20. cos1315cos(5)sin1315sin(5)Cofunction Identities Prove the cofunction identity using the Addition and Subtraction Formulas. 21. tan(2u)=cotu Cofunction Identities Prove the cofunction identity using the Addition and Subtraction Formulas. 22. cot(2u)=tanu 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E Proving Identities Prove the identity. 33. tan(x+3)=3+tanx13tanx 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E Expressions Involving Inverse Trigonometric Functions Write the given expression in terms of x and y only. 48. tan(sin1 x + cos1 y)49E 50E 51E 52E 53E 54E 55E Evaluating Expressions Involving Trigonometric Functions Evaluate each expression under the given conditions. 56. sin( ); tan=43, in Quadrant III, sin=10/10, in Quadrant IV 57E Evaluating Expressions Involving Trigonometric Functions Evaluate each expression under the given conditions. 58. tan( + ); cos=13, in Quadrant III, sin=14, in Quadrant II Expressions in Terms of Sine Write the expression in terms of sine only. 59. 3sinx+cosx 60E 61E 62E 63E 64E Difference Quotient Let f(x) = cos x and g(x) = sin x. Use Addition or subtraction Formulas to show the following. 65. f(x+h)f(x)h=cosx(1coshh)sinx(sinhh)66E 67E 68E 69E Sum of Two Angles Refer to the figure. Show that + = , and find tan .71E 72E Angle Between Two Lines In this exercise we find a formula for the angle formed by two lines in a coordinate plane. (a) If L is a line in the plane and is the angle formed by the line and the x-axis as shown in the figure, show that the slope m of the line is given by mtanx (b) Let L1 and L2 be two nonparallel lines in the plane with slopes m1 and m2, respectively. Let be the acute angle formed by the two lines (see the following figure). Show that tan=m2m11+m1m2 (c) Find the acute angle formed by the two lines y=13x+1andy=12x3 (d) Show that if two lines are perpendicular, then the slope of one is the negative reciprocal of the slope of the other. [Hint: First find an expression for cot .]FindA+B+Cin the figure. [Hint: First use an Addition Formula to find tan(A + B).]75E Interference Two identical tuning forks are struck, one a fraction of a second after the other. The sounds produced are modeled by f1(t) = C sin t and f2(t) = C sin(t + ). The two sound waves interfere to produce a single sound modeled by the sum of these functions f(t)=Csint+Csin(t+) (a) Use the Addition Formula for Sine to show that f can be written in the form f(t) = A sin t + B cos t, where A and B are constants that depend on . (b) Suppose that C = 10 and = /3. Find constants k and so that f(t) = k sin(t + ).PROVE: Addition Formula for Sine In the text we proved only the Addition and Subtraction Formulas for Cosine. Use these formulas and the cofunction identities sinx=cos(2x)cosx=sin(2x) to prove the Addition Formula for Sine. [Hint: To get started, use the first cofunction identity to write sin(s+t)=cos(2(s+t))=cos((2s)t) and use the Subtraction Formula for Cosine.]78E If we know the values of sin x and cos x, we can find the value of sin 2x by using the _____ Formula for Sine. State the formula: sin 2x = __________.If we know the value of cos x and the quadrant in which x/2 lies, we can find the value of sin(x/2) by using the _____ Formula for Sine. State the formula: sin(x/2) = __________.3E

Page: [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]