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mathematics
calculus
Questions and Answers of
Calculus
In Exercises 1-4, sketch the graph of the function and compare the graph to the graph of the parent inverse trigonometric function. 1. y = 2 arcsin x 2. f (x) =arctan 2x 3. f (x) = π / 2 + arctan x
In Exercises 1-4, use a graphing utility to graph the function. 1. f (x) = 2 arccos 2x 2. f (x) = π arcsin 4x 3. f (x) = arctan(2x − 3) 4. f (x) = −3 + arctan πx
In Exercises 1 and 4, write the function in terms of the sine function by using the identity A cos wt + B sin wt = √A2 + B2 sin (wt + arctan A/B). Use a graphing utility to graph both forms of the
In Exercises 1–4, fill in the blank. If not possible, state the reason. (Note: The notation x → c+ indicates that x approaches c from the right and x → c indicates that x approaches c
A boat is pulled in by means of a winch located on a dock 5 feet above the deck of the boat (see figure). Let θ be the angle of elevation from the boat to the winch and let s be the
1. A ________ measures the acute angle that a path or line of sight makes with a fixed north-south line. 2. A point that moves on a coordinate line is in simple ________ ________ when its distance d
In Exercises 1-4, find the altitude of the isosceles triangle shown in the figure. Round your answers to two decimal places. 1. θ = 45°, b = 6 2. θ = 22°, b = 14 3. θ = 32°, b = 8 4. θ =
The sun is 25° above the horizon. Find the length of a shadow cast by a building that is 100 feet tall (see figure).
The sun is 20° above the horizon. Find the length of a shadow cast by a park statue that is 12feettall.
A ladder that is 20 feet long leans against the side of a house. The angle of elevation of the ladder is 80°. Find the height from the top of the ladder to the ground.
The length of a shadow of a tree is 125feet when the angle of elevation of the sun is 33°. Approximate the height of the tree.
At a point 50 feet from the base of a church, the angles of elevation to the bottom of the steeple and the top of the steeple are 35° and 48°, respectively. Find the height of the steeple.
An observer in a lighthouse 350 feet above sea level observes two ships directly offshore. The angles of depression to the ships are 4° and 6.5° (see figure). How far apart are the ships?
A passenger in an airplane at an altitude of 10kilometers sees two towns directly to the east of the plane. The angles of depression to the towns are 28° and 55° (see figure). How far apart
The height of an outdoor basketball backboard is 12 / 1 / 2 feet, and the backboard casts a shadow 17 feet long. (a) Draw a right triangle that gives a visual representation of the problem. Label the
An engineer designs a 75-foot cellular telephone tower. Find the angle of elevation to the top of the tower at a point on level ground 50 feet from its base.
A cellular telephone tower that is 120feet tall is placed on top of a mountain that is 1200feet above sea level. What is the angle of depression from the top of the tower to a cell phone user who
A Global Positioning System satellite orbits 12,500 miles above Earth's surface (see figure). Find the angle of depression from the satellite to the horizon. Assume the radius of Earth is 4000 miles.
You are holding one of the tethers attached to the top of a giant character balloon that is floatingapproximately 20 feet above ground level. You are standing approximately 100 feet ahead of the
You observe a plane approaching overhead and assume that its speed is 550 miles per hour. The angle of elevation of the plane is 16° at one time and 57° one minute later. Approximate the altitude
The designers of a water park have sketched a preliminary drawing of a new slide (see figure).(a) Find the height h of the slide. (b) Find the angle of depression Î from the top of the
A police department has set up a speed enforcement zone on a straight length of highway. A patrol car is parked parallel to the zone, 200feet from one end and 150feet from the other end (see
During takeoff, an airplane's angle of ascent is 18° and its speed is 260feet per second. (a) Find the plane's altitude after 1 minute. (b) How long will it take for the plane to climb to an
An airplane flying at 550 miles per hour has a bearing of 52°. After flying for 1.5 hours, how far north and how far east will the plane have traveled from its point of departure?
A jet leaves Reno, Nevada, and heads toward Miami, Florida, at a bearing of 100°. The distance between the two cities is approximately 2472miles. (a) How far north and how far west is Reno relative
A ship leaves port at noon and has a bearing of S 29° W. The ship sails at 20 knots. (a) How many nautical miles south and how many nautical miles west will the ship have traveled by 6:00P.M.? (b)
A privately owned yacht leaves a dock in Myrtle Beach, South Carolina, and heads toward Freeport in the Bahamas at a bearing of S 1.4° E. The yacht averages a speed of 20 knots over the
A ship is 45 miles east and 30 miles south of port. The captain wants to sail directly to port. What bearing should the captain take?
An airplane is 160 miles north and 85 miles east of an airport. The pilot wants to fly directly to the airport. What bearing should the pilottake?
A surveyor wants to find the distance across a pond (see figure). The bearing from A to B is N 32° W. The surveyor walks 50 meters from A to C, and at the point C the bearing to B is N 68°
Fire tower A is 30 kilometers due west of fire tower B. A fire is spotted from the towers, and the bearings from A and B are N 76° E and N 56° W, respectively (see figure). Find the distance
Determine the angle between the diagonal of a cube and the diagonal of its base, as shown in the figure.
Determine the angle between the diagonal of a cube and its edge, as shown in the figure.
Find the length of the sides of a regular pentagon inscribed in a circle of radius 25 inches.
Find the length of the sides of a regular hexagon inscribed in a circle of radius 25 inches.
In Exercises 1-4, find a model for simple harmonic motion satisfying the specified conditions. Displacement (t = 0) Amplitude Period 1. 0 4 centimeters 2 seconds 2. 0 3
A point on the end of a tuning fork moves in simple harmonic motion described by d = a sin t. Find ω given that the tuning fork for middle C has a frequency of 262 vibrations per second.
In Exercises 1-4, solve the right triangle shown in the figure for all unknown sides and angles. Round your answers to two decimal places 1. A = 60°, c = 12 2. B = 25°, b = 4 3. B = 72.8°, a =
A buoy oscillates in simple harmonic motion as waves go past. The buoy moves a total of 3.5feet from its low point to its high point (see figure), and it returns to its high point every 10 seconds.
In Exercises 1-2, for the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b)the frequency, (c)the value of d when t = 5, and (d) the least
A ball that is bobbing up and down on the end of a spring has a maximum displacement of 3 inches. Its motion (in ideal conditions) is modeled by y = 1/4 cos 16t, t > 0, where y is measured in feet
The numbers of hours H of daylight in Denver, Colorado, on the 15th of each month starting with January are: 9.68, 10.72, 11.92, 13.25, 14.35, 14.97, 14.72, 13.73, 12.47, 11.18, 10.00, and 9.37. A
The table shows the average sales S (in millions of dollars) of an outerwear manufacturer for each month t, where t = 1 corresponds to January.(a) Create a scatter plot of the data. (b) Find a
In Exercises 59 and 60, determine whether the statement is true or false. Justify your answer. 1. The Leaning Tower of Pisa is not vertical, but when you know the angle of elevation Θ to the top of
Estimate the number of degrees in the angle.1.2.
Use a graphing utility to graph the function. 1. f (x) = 2 arcsin(x / 2) 2. f (x) = 3 arccos x 3. f (x) = arctan(x / 2) 4. f (x) = −arcsin 2x
Find the exact value of the expression. 1. cos (arctan 3 / 4) 2. tan (arcos 3 / 5) 3. sec (arctan 12 / 5) 4. cot [ arcsin (12 / 13)]
Convert the degree measure to radian measure. Round to three decimal places. 1. 4500 2. 1200 3. -160 4. -1120 5. 20.360 6. 45.140 7. -8.560 8. -300.120
Write an algebraic expression that is equivalent to the given expression. 1. tan [arccos (x / 2)] 2. sec [arcsin (x − 1)]
1. The height of a radio transmission tower is 70 meters, and it casts a shadow of length 30 meters. Draw a right triangle that gives a visual representation of the problem. Label the known and
1. A ship leaves port at noon and has a bearing of N 45° E. The ship sails at 15 knots. How many nautical miles north and how many nautical miles east will the ship have traveled by 4:00 P.M? 2. A
Determine whether the statement is true or false. Justify your answer. 1. y = sin θ is not a function because sin 30° = sin 150°. 2. Because tan (3π / 4) = -1, arctan (-1) = 3π /4.
1. Describe the behavior of f (θ) = sec θ at the zeros of g(θ) = cos θ. Explain.2. (a) Use a graphing utility to complete the table.(b) Make a
1. When graphing the sine and cosine functions, determining the amplitude is part of the analysis. Explain why this is not true for the other four trigonometric functions.2. A weight is suspended
The base of the triangle shown in the figure is also the radius of a circular arc.(a) Find the area A of the shaded region as a function of θ for 0 (b) Use a graphing utility to graph
Convert the radian measure to degree measure. Round to three decimal places, if necessary. 1. 3π / 10 2. 7π / 5 3. -3π / 5 4. -11π /6 5. 5.2 6. 7 7. -2.15 8. -4.63
1. Find the length of the arc on a circle with a radius of 20 inches intercepted by a central angle of 138° 2. At what speed is a bicycle traveling when its 29-inch-diameter tires are rotating at an
Find the area of the sector of a circle of radius r and central angle θ. Radius........................... Central Angle θ 1. 20 inches........................... 1500 2. 7.5 millimeters
(a) Sketch the angle in standard position, (b) Determine the quadrant in which the angle lies, and (c) Determine two coterminal angles (one positive and one negative). 1. 850 2. 3100 3. -1100 4.
Find the exact values of the six trigonometric functions of the angle θ.1.2.
Use a calculator to evaluate each function. Round your answers to four decimal places. (Be sure the calculator is in the correct mode.) 1. (a) cos 38.90............ (b) sec 79.30 2. (a) sin (π / 18)
Use the given function value and the trigonometric identities to find the exact value of each indicated trigonometric function. 1. sin θ = 1/3 (a) csc θ ..........................................
1. A train travels 3.5 kilometers on a straight track with a grade of 1.2° (see figure). What is the vertical rise of the train in that distance?2. A guy wire runs from the ground to the top of a
The point is on the terminal side of an angle in standard position. Find the exact values of the six trigonometric functions of the angle. 1. (12, 16) 2. (7, -24) 3. (-0.5, 4.5)
Find the exact values of the remaining five trigonometric functions of satisfying the given conditions. 1. sec θ 6 /5, tan θ < 0 2. csc θ = 3 / 2, cos θ < 0 3. tan θ = 7 / 3, cos θ < 0
Find the reference angle θ'. Then sketch in standard position and label θ'. 1. θ = 2640 2. θ = 6350 3. θ = -6π / 5 4. θ = 17π / 3
Evaluate the sine, cosine, and tangent of the angle without using a calculator. 1. 4950 2. 1200 3. -1500 4. -4200
Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is in the correct mode.) 1. sin 1060 2. tan 370 3. tan (-17π / 15) 4. cos
Find the point (x, y) on the unit circle that corresponds to the real number t. Use the result to evaluate sin t, cos t, and tan t. 1. t = 2π / 3 2. t = 7π / 4 3. t = 7π / 6
Sketch the graph of the function. (Include two full periods.) 1. y = sin 6x 2. y = -cos 3x 3. f (x) = 5 sin(2x / 5) 4. f (x) = 8 cos ( -x / 4) 5. y = 5 + sin πx
Sound waves can be modeled using sine functions of the form y = a sin bx, where x is measured in seconds. (a) Write an equation of a sound wave whose amplitude is 2 and whose period is 1/264
The times S of sunset (Greenwich Mean Time) at 40° north latitude on the 15th of each month starting with January are: 16:59, 17:35, 18:06, 18:38, 19:08, 19:30, 19:28, 18:57, 18:10, 17:21, 16:44,
Sketch the graph of the function. (Include two full periods.) 1. f (x) = 3 tan 2x 2. f (t) = tan(t + π/2) 3. f (x) = 1/2 cot x 4. g(t) = 2 cot 2t 5. f (x) = 3 sec x
Use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as x increases without bound. 1. f (x) = x cos
Find the exact value of the expression. 1. arcsin (-1 / 2) 2. arcsin (-1) 3. arcos (-√2 / 2) 4. arccos √2 / 2 5. cos-1 1 6. cos-1 √3 / 2
Use a calculator to approximate the value of the expression, if possible. Round your result to two decimal places. 1. arcsin 3.26 2. arcsin (-0.363) 3. sin-1 (0.17) 4. sin-1 0.15 5. arcos 0.372 6.
1. sin u / cos y = __________ 2. 1 / sin u = _______ 3. 1 / tan u = ______ 4. sec (π / 2 - u) = ______ 5. sin2 u + cos2 u = ______ 6. sin(- u) = _____
In Exercises 1-4, match the trigonometric expression with its simplified form. (a) csc x (b) -1 (c) 1 (d) sin x tan x (e) sec2 x (f) sec x 1. sec x cos x 2. cot2 x − csc2 x 3. cos x(1 + tan2
In Exercises 1-4, use the fundamental identities to simplify the expression. (There is more than one correct form of each answer). 1. tan Θ cot Θ / sec Θ 2. cos (π / 2 - x )sec x 3. tan2 x -
In Exercises 1-4, factor the expression. Use the fundamental identities to simplify, if necessary. (There is more than one correct form of each answer.) 1. sec2 x - 1 / sec x - 1 2. cos x - 2 / cos2
In Exercises 1-4, use the fundamental identities to simplify the expression. (There is more than one correct form of each answer.) 1. tan Θ csc Θ 2. tan(−x) cos x 3. sin ϕ(csc ϕ - sin ϕ) 4.
In Exercises 1 and 2, perform the multiplication and use the fundamental identities to simplify. (There is more than one correct form of each answer.) 1. (sin x + cos x)2 2. (2 csc x + 2)(2 csc x −
In Exercises 1-4, perform the addition or subtraction and use the fundamental identities to simplify. (There is more than one correct form of each answer.) 1. 1 / 1 + cos x + 1 / 1 - cos x 2. 1 /
In Exercises 1 and 2, rewrite the expression so that it is not in fractional form. (There is more than one correct form of each answer.) 1. sin2 y / 1 - cos y 2. 5 / tan x + sec x
In Exercises 1 and 2, use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically. 1. tan x + 1 / sec x + csc x 2. 1 /
In Exercises 1-4, use the trigonometric substitution to write the algebraic expression as a trigonometric function of Θ, where 0 < Θ < π / 2. 1. √9 - x2, x = 3 cos Θ 2. √49 - x2, x = 7 sin
In Exercises 1 and 2, use the trigonometric substitution to write the algebraic equation as a trigonometric equation of , where - π / 2 < Θ < π / 2. Then find sin Θ and cos Θ. 1. √2 = √4
In Exercises 1 and 2, use a graphing utility to solve the equation for, where 0 ≤ Θ < 2π. 1. sin Θ = √1 - cos2 Θ 2. sec Θ = √1 + tan2 Θ
In Exercises 1-4, rewrite the expression as a single logarithm and simplify the result. 1. In│sinx│ + In│cot x│ 2. In │cos x│ - In│sin x│ 3. In│tan t│ - In(1 - cos2 t) 4. In(cos2
The forces acting on an object weighing W units on an inclined plane positioned at an angle of Î with the horizontal (see figure) are modeled by µ W cos Î = W sin
The rate of change of the function f (x) = sec x + cos x is given by the expression sec x tan x − sin x. Show that this expression can also be written as sin x tan2 x.
In Exercises 1 and 2, determine whether the statement is true or false. Justify your answer. 1. The quotient identities and reciprocal identities can be used to write any trigonometric function in
In Exercises 1 and 2, fill in the blanks. (Note: The notation x → c+ indicates that x approaches c from the right and x → c - indicates that x approaches c from the left.)
In Exercises 1-4, use the given conditions to find the values of all six trigonometric functions. 1. sec x = - 5 / 2, tan x < 0 2. csc x = - 7 / 6, tan x > 0 3. sin Θ = - 3 / 4, cos Θ > 0 4. cos
Describe the error. Sin Θ / cos(- Θ) = sin Θ / - cos Θ = - tan Θ
Use the trigonometric substitution u = a tan Θ, where - π / 2 < Θ < π / 2 and a > 0, to simplify the expression √a2 + u2.
Write each of the other trigonometric functions of Θ in terms of sin Θ.
Rewrite the expression below in terms of sin Θ and cos Θ. sec Θ(1 + tan Θ) / sec Θ + csc Θ
1. Write each of the other trigonometric functions of in terms of cos θ. 2. Verify that for all integers n, cos[(2n + 1) π / 2] = 0.
The number of hours of daylight that occur at any location on Earth depends on the time of year and the latitude of the location. The equations below model the numbers of hours of daylight in Seward,
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