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Algebra And Trigonometry 10th Edition Ron Larson - Solutions

In Exercises 1-4, verify the identity. 1. cos x(tan2 x + 1) = sec x 2. sec2 x cot x - cot x = tan x 3. sin(π / 2 - Θ) tan Θ = sin Θ 4. cot(π / 2 - π) csc x = sec x
In Exercises 1-4, solve the equation. 1. sin x = √3 - sin x 2. 4 cos Θ = 1 + 2 cos Θ 3. 3 √3 tan u = 3 4. 1 / 2 sec x - 1 = 0
In Exercises 1-4, find all solutions of the equation in the interval [0,2π). 1. sin3 x = sin x 2. 2 cos2 x + 3 cos x = 0 3. cos2 x + sin x = 1 4. sin2 x + 2 cos x =
In Exercises 1-4, solve the equation. 1. tan2 x − 2 tan x = 0 2. 2 tan2 x − 3 tan x = −1 3. tan2 Θ + tan Θ − 6 = 0 4. sec2 x + 6 tan x + 4 = 0
In Exercises 1-2, find the exact values of the sine, cosine, and tangent of the angle. 1. 75o = 120o - 45o 2. 375o = 135o + 240o
In Exercises 1 and 2, use the given conditions and fundamental trigonometric identities to find the values of all six trigonometric functions. 1. cos Θ = - 2 / 5, tan Θ > 0 2. cot x = - 2 / 3, cos x < 0
Exercises 1 and 2, write the expression as the sine, cosine, or tangent of an angle. 1. sin 60o cos 45o - cos 60o sin 45o 2. tan 68o - tan 115o / 1 + tan 68o tan 115o
In Exercises 1-4, find the exact value of the trigonometric expression given that tan u = 3 / 4 and cos v = - 4 / 5.( u is in Quadrant I and v is in Quadrant III). 1. sin (y + v) 2. tan (u + v) 3. cos (u - v) 4. sin (u - v)
Verify the identity. 1. Cos (x + π / 2) = - sin x 2. Tan (x - π / 2) = - cos x 3. Tan (π - x) = - tan x 4. Sin (x - π) = - sin x
Find all solutions of the equation in the interval [0, 2π. 1. Sin (x + π / 4) - sin (x - π / 4) = 1 2. Cos (x + π / 6) - cos (x - π / 6) = 1
Use the given conditions to find the exact values of sin 2u, cos 2u, and tan 2u using the double-angle formulas. Sin u = 4 / 5, 0 < u π / 2
Use the double-angle formulas to verify the identity algebraically and use a graphing utility to confirm your result graphically. Sin 4x = 8 cos3 x sin x - 4 cos x sin x
Use the power-reducing formulas to rewrite the expression in terms of first powers of the cosines of multiple angles. 1. Tan2 3x 2. Sin2 x cos2 x
Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. 1. - 75o 2. 5π / 12
In Exercises 1-4, use the fundamental trigonometric identities to simplify the expression. (There is more than one correct form of each answer.) 1. 1 / cot2 x + 1 2. tan Θ / 1 - cos2 Θ 3. tan2 x(csc2 x - 1) 4. cot2 x(sin2 x)
Use the given conditions to(a) Determine the quadrant in which u / 2 lies, and(b) Find the exact values of Sin (u / 2), cos (u / 2), and tan ( u / 2) using the half-angle formulas.1. Tan u = 4 / 3, π < u < 3π / 22. Sin u = 3 / 5, 0 < u < π / 2
Use the product-to-sum formulas to rewrite the product as a sum or difference. 1. Cos 4 θ Sin 6 θ 2. 2 Sin 7 θ Cos 3 θ
Use the sum-to-product formulas to rewrite the sum or difference as a product. 1. Cos 6 θ + Cos 5 θ 2. Sin 3x − Sin x
A baseball leaves the hand of a player at first base at an angle of with the horizontal and at an initial velocity of v0 = 80 feet per second. A player at second base 100 feet away catches the ball. Find when the range r of a projectile is r = 1 / 32 vo2 sin 2 θ
A trough for feeding cattle is 4 meters long and its cross sections are isosceles triangles with the two equal sides being 1/2 meter (see figure). The angle between the two sides is Î¸.(a) Write the volume of the trough as a function of Î¸ / 2. (b) Write the volume of the
Determine whether the statement is true or false. Justify your answer. 1. If π / 2 < θ < π, then θ / 2 < 0. 2. Cot x sin2 x = cos x sin x 3. 4 sin (- x) cos (- x) = - 2 sin 2x 4. 4 sin 45o cos 15o = 1 + √3
Is it possible for a trigonometric equation that is not an identity to have an infinite number of solutions? Explain.
1. An ________ triangle is a triangle that has no right angle. 2. For triangle ABC, the Law of Sines is a / sin A = ________ = c / sin C. 3. Two ________ and one ________ determine a unique triangle. 4. The area of an oblique triangle ABC is 1 / 2 bc sin A = 1 / 2 ab sin C = ________.
Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. 1. A = 110°, a = 125, b = 100 2. A = 110°, a = 125, b = 200 3. A = 76°, a = 18, b = 20 4. A = 76°, a = 34, b = 21 5. A = 58°, a = 11.4, b = 12.8
Find values for b such that the triangle has (a) one solution, (b) two solutions (if possible), and (c) no solution. 1. A = 36°, a = 5 2. A = 60°, a = 10 3. A = 105°, a = 80 4. A = 132°, a = 215
Find the area of the triangle. Round your answers to one decimal place. 1. A = 125°, b = 9, c = 6 2. C = 150°, a = 17, b = 10 3. B = 39°, a = 25, c = 12 4. A = 72°, b = 31, c = 44 5. C = 103° 15', a = 16, b = 28
A tree grows at an angle of 4° from the vertical due to prevailing winds. At a point 40meters from the base of the tree, the angle of elevation to the top of the tree is 30° (see figure).(a) Write an equation that you can use to find the height h of the tree.(b) Find the height of the tree.
A boat is traveling due east parallel to the shoreline at a speed of 10 miles per hour. At a given time, the bearing to a lighthouse is S 70° E, and 15 minutes later the bearing is S 63° E (see figure). The lighthouse is located at the shoreline. What is the distance from the boat to the
The bearing from the Pine Knob fire tower to the Colt Station fire tower is N 65° E, and the two towers are 30 kilometers apart. A fire spotted by rangers in each tower has a bearing of N 80° E from Pine Knob and S 70° E from Colt Station (see figure). Find the distance of the fire from
A bridge is built across a small lake from a gazebo to a dock (see figure). The bearing from the gazebo to the dock is S 41° W. From a tree 100meters from the gazebo, the bearings to the gazebo and the dock are S 74° E and S 28° E, respectively. Find the distance from the gazebo to the
1. A 10-meter utility pole casts a 17-meter shadow directly down a slope when the angle of elevation of the sun is 42° (see figure). Find Î¸, the angle of elevation of the ground.2. A plane flies 500 kilometers with a bearing of 316° from Naples to Elgin (see figure). The plane
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.1.2. 3. 4. 5. A = 102.4°, C = 16.7°, a = 21.6 6. A = 24.3°, C = 54.6°, c = 2.68 7. A = 830 20', C = 54.60, c = 18.1 8. A = 50 40', B = 80 15', b= 4.8
1. The angles of elevation to an airplane from two points A and B on level ground are 55° and 72°, respectively. The points A and B are 2.2 miles apart, and the airplane is east of both points in the same vertical plane. (a) Draw a diagram that represents the problem. Show the known quantities on
Air traffic controllers continuously monitor the angles of elevation Î¸ and ˆ… to an airplane from an airport control tower and from an observation post 2 miles away (see figure). Write an equation giving the distance d between the plane and the observation post in terms of
In the figure, Î± and Î² are positive angles.(a) Write Î± as a function of Î². (b) Use a graphing utility to graph the function in part (a). Determine its domain and range. (c) Use the result of part (a) to write c as a function of Î². (d)
Determine whether the statement is true or false. Justify your answer. 1. If a triangle contains an obtuse angle, then it must be oblique. 2. Two angles and one side of a triangle do not necessarily determine a unique triangle. 3. When you know the three angles of an oblique triangle, you can solve
Describe the error.1. The area of the triangle with C = 58°, b = 11 feet, and c = 16 feet is Area = 1/2 (11) (16) (sin 580) = 88 (sin 580) ‰ˆ 74.63 square feet.2. In the figure, a triangle is to be formed by drawing a line segment of length a from (4, 3) to the positive x-axis. For
Can the Law of Sines be used to solve a right triangle? If so, use the Law of Sines to solve the triangle with B = 500, C = 900, a = 10. Is there another way to solve the triangle? Explain.
(a) Write the area A of the shaded region in the figure as a function of Î¸.(b) Use a graphing utility to graph the function.(c) Determine the domain of the function. Explain how decreasing the length of the eight-centimeter line segment affects the area of the region and the domain of
In the figure, a beam of light is directed at the blue mirror, reflected to the red mirror, and then reflected back to the blue mirror. Find PT, the distance that the light travels from the red mirror back to the blue mirror.
A skydiver falls at a constant downward velocity of 120 miles per hour. In the figure, vector u represents the skydiver's velocity. A steady breeze pushes the skydiver to the east at 40 miles per hour. Vector v represents the wind velocity.(a) Write the vectors u and v in component form. (b) Let s
The vectors u and v have the same magnitudes in the two figures. In which figure is the magnitude of the sum greater? Explain(a)(b)
Four basic forces are in action during flight: weight, lift, thrust, and drag. To fly through the air, an object must overcome its own weight. To do this, it must create an upward force called lift. To generate lift, a forward motion called thrust is needed. The thrust must be great enough to
A triathlete sets a course to swim S 25° E from a point on shore to a buoy 3 / 4 mile away. After swimming 300 yards through a strong current, the triathlete is off course at a bearing of S 35° E. Find the bearing and distance the triathlete needs to swim to correct her course.
A group of hikers is lost in a national park. Two ranger stations receive an emergency SOS signal from the hikers. Station B is 75 miles due east of station A. The bearing from station A to the signal is S 60° E and the bearing from station B to the signal is S 75° W. (a) Draw a diagram that
You are seeding a triangular courtyard. One side of the courtyard is 52 feet long and another side is 46 feet long. The angle opposite the 52-foot side is 65°. (a) Draw a diagram that gives a visual representation of the problem. (b) How long is the third side of the courtyard? (c) One bag of
For each pair of vectors, find the value of each expression. (i) ∥u∥ (ii) ∥v∥ (iii) ∥u + v∥ (iv) ∥ u / ∥u∥∥ (v) ∥ v / ∥v∥ ∥ (vi) ∥ u + v / ∥ u + v∥ ∥ (a) u = 〈 1 , - 1〉 v = 〈-1, 2〉 (b) u = 〈0, 1〉 v = 〈3, -3〉 (c) u = 〈1, 1 / 2〉 v = 〈2,
Write the vector w in terms of u and v, given that the terminal point of w bisects the line segment (see figure).
Prove that if u is orthogonal to v and w, then u is orthogonal to cv + dw for any scalars c and d.
Two forces of the same magnitude F1 and F2 act at angles θ1 and θ2, respectively. Use a diagram to compare the work done by F1 with the work done by F2 in moving along the vector P̅Q̅ when (a) θ1 = - θ2 (b) θ1 - 600 and θ2 - 300.
For each graph of the roots of a complex number, write each of the roots in trigonometric form.(a)(b)
1. The standard form of the Law of Cosines for cos B = a2 + c2 − b2 / 2ac is ________. 2. When solving an oblique triangle given three sides, use the ________ form of the Law of Cosines to solve for an angle. 3. When solving an oblique triangle given two sides and their included angle, use the
Find the missing values by solving the parallelogram shown in the figure. (The lengths of the diagonals are given by c and d.)1. a =5, b = 8, Î¸ = 450 2. a = 25, b =35, ˆ… = 1200 3. a =10, b = 14, c = 20
Determine whether the Law of Cosines is needed to solve the triangle. Then solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. 1. a = 8, c = 5, B = 40° 2. a = 10, b = 12, C = 70° 3. A = 24°, a = 4, b = 18
Use Heron's Area Formula to find the area of the triangle. 1. a = 6, b = 12, c = 17 2. a = 33, b = 36, c = 21 3. a = 2.5, b = 10.2, c = 8 4. a = 12.32, b = 8.46, c = 15.9 5. a = 1, b = 1 / 2, c = 5 / 4
1. To approximate the length of a marsh, a surveyor walks 250 meters from point A to point B, then turns 75° and walks 220 meters to point C (see figure). Approximate the length AC of the marsh.2. Determine the angle Î¸ in the design of the streetlight shown in the figure.
1. A baseball player in center field is approximately 330 feet from a television camera that is behind home plate. A batter hits a fly ball that goes to the wall 420 feet from the camera (see figure). The camera turns 8° to follow the play. Approximately how far does the center fielder have to
A 100-foot vertical tower is built on the side of a hill that makes a 6° angle with the horizontal (see figure). Find the length of each of the two guy wires that are anchored 75 feet uphill and downhill from the base of the tower.
Use the Law of Cosines to solve the triangle. Round your answers to two decimal places.1.2. 3. 4. 5.
On a map, Minneapolis is 165millimeters due west of Albany, Phoenix is 216 millimeters from Minneapolis, and Phoenix is 368 millimeters from Albany (see figure).(a) Find the bearing of Minneapolis from Phoenix. (b) Find the bearing of Albany from Phoenix.
A boat race runs along a triangular course marked by buoys A, B, and C. The race starts with the boats headed west for 3700 meters. The other two sides of the course lie to the north of the first side, and their lengths are 1700 meters and 3000 meters. Draw a diagram that gives a visual
A plane flies 810 miles from Franklin to Centerville with a bearing of 75°. Then it flies 648 miles from Centerville to Rosemount with a bearing of 32°. Draw a diagram that gives a visual representation of the problem. Then find the straight-line distance and bearing from Franklin to Rosemount.
1. A triangular parcel of land has 115meters of frontage, and the other boundaries have lengths of 76 meters and 92 meters. What angles does the frontage make with the two other boundaries? 2. A triangular parcel of ground has sides of lengths 725 feet, 650 feet, and 575 feet. Find the measure of
Two ships leave a port at 9 A.M. One travels at a bearing of N 53° W at 12 miles per hour, and the other travels at a bearing of S 67° W at s miles per hour. (a) Use the Law of Cosines to write an equation that relates s and the distance d between the two ships at noon. (b) Find the speed s that
An engine has a seven-inch connecting rod fastened to a crank (see figure).(a) Use the Law of Cosines to write an equation giving the relationship between x and Î¸. (b) Write x as a function of Î¸. (Select the sign that yields positive values of x.) (c) Use a graphing
1. A triangular parcel of land has sides of lengths 200 feet, 500 feet, and 600 feet. Find the area of the parcel.2. A parking lot has the shape of a parallelogram (see figure). The lengths of two adjacent sides are 70 meters and 100 meters. The angle between the two sides is 70°. What is the
1. You want to buy a triangular lot measuring 510 yards by 840 yards by 1120 yards. The price of the land is $2000 per acre. How much does the land cost? (Hint: 1 acre = 4840 square yards) 2. You want to buy a triangular lot measuring 1350 feet by 1860 feet by 2490 feet. The price of the land is
Determine whether the statement is true or false. Justify your answer. 1. In Heron's Area Formula, s is the average of the lengths of the three sides of the triangle. 2. In addition to SSS and SAS, the Law of Cosines can be used to solve triangles with AAS conditions. 3. What familiar formula do
Describe how the Law of Cosines can be used to solve the ambiguous case of the oblique triangle ABC, where a = 12 feet, b = 30 feet, and A = 20°. Is the result the same as when the Law of Sines is used to solve the triangle? Describe the advantages and the disadvantages of each method.
The Law of Cosines was used to solve a triangle in the two-solution case of SSA. Can the Law of Cosines be used to solve the no-solution and single-solution cases of SSA? Explain.
1. To solve the triangle, would you begin by using the Law of Sines or the Law of Cosines? Explain.(a)(b) 2. Use the Law of Cosines to prove each identity. (a) 1/2 bc (1 +cos A) = a + b + c / 2 ˆ™ -a +b +c / 2 (b) 1/2 bc (1 - cos A) = a - b + c / 2 ˆ™ a + b + c / 2
1. You can use a ________ ________ ________ to represent a quantity that involves both magnitude and direction. 2. The directed line segment PQ \ has ________ point P and ________ point Q. 3. The set of all directed line segments that are equivalent to a given directed line segment PQ \ is a
Write a program for your graphing utility that graphs two vectors and their difference given the vectors in component form.
Use the program in Exercise 102 to find the difference of the vectors shown in the figure.1.2.
Consider two forces F1 = {10, 0} and F2 = 5{cos θ , sin θ}. (a) Find ∥F1 + F2∥ as a function of θ. (b) Use a graphing utility to graph the function in part (a) for 0 ≤ θ < 2π. (c) Use the graph in part (b) to determine the range of the function. What is its maximum, and for what
Use the figure to determine whether each statement is true or false. Justify your answer.(a) a = ˆ’d (b) c = s (c) a + u = c (d) v + w = ˆ’s (e) a + w = ˆ’2d (f) a + d = 0 (g) u ˆ’ v = ˆ’2(b + t) (h) t ˆ’ w = b - a
Give geometric descriptions of (a) Vector addition (b) Scalar multiplication.
Identify the quantity as a scalar or as a vector. Explain. (a) The muzzle velocity of a bullet (b) The price of a company's stock (c) The air temperature in a room (d) The weight of an automobile
Find the component form and magnitude of the vector v.1.2. 3.
Use the figure to sketch a graph of the specified vector. To print an enlarged copy of the graph, go to MathGraphs.com.1. -v 2. 5v 3. u + v 4. u + 2v 5. u - v 6. v - ½u
Find (a) u + v, (b) u - v, (c) 2u - 3v. Then sketch each resultant vector. 1. u = (2, 1), v = (1, 3) 2. u = (2, 3), v = (4, 0)
Find the magnitude of the scalar multiple, where u = (2, 0) and v = (3, 6). 1. ∥5u∥ 2. ∥4v∥ 3. ∥-3v∥ 4. ∥-3/4u∥
Find a unit vector u in the direction of v. Verify that ∥u∥ = 1. 1. v = (3, 0) 2. v = (0, −2) 3. v = (−2, 2)
Find the vector v with the given magnitude and the same direction as u. 1. ∥v∥ = 10, u = (-3, 4) 2. ∥v∥ = 3, u = (-12, -5) 3. ∥v∥ = 9, u = (2, 5) 4. ∥v∥ = 8, u = (3, 3)
The initial and terminal points of a vector are given. Write the vector as a linear combination of the standard unit vectors i and j. Initial Point .......................................................... Terminal Point 1. (−2, 1) .............................................................
Find the component form of v and sketch the specified vector operations geometrically, where u = 2i - j and w = i + 2j. 1. v = 3/2u 2. v = 3/4w 3. v = u + 2w 4. v = −u + w
Find the magnitude and direction angle of the vector v. 1. v = 6i − 6j 2. v = −5i + 4j 3. v = 3(cos 60°i + sin 60°j) 4. v = 8(cos 135°i + sin 135°j)
Find the component form of v given its magnitude and the angle it makes with the positive x-axis. Then sketch v. Magnitude ............................................................................... Angle 1. ∥v∥ = 3
Find the component form of the sum of u and v with direction angles u and v. 1. ∥u∥ = 4, θu = 60o ∥v∥ = 4, θv = 90o 2. ∥u∥ = 20, θu = 45o ∥v∥ = 50, θv = 180o
Use the Law of Cosines to find the angle between the vectors. Assume 0 180. 1. v = i + j, w = 2i − 2j 2. v = i + 2j, w = 2i − j
Find the angle between the forces given the magnitude of their resultant. (Hint: Write force 1 as a vector in the direction of the positive x-axis and force 2 as a vector at an angle with the positive x-axis.) Force 1 ........................ Force 2 ........................ Resultant Force 1. 45
1. A gun with a muzzle velocity of 1200feet per second is fired at an angle of 6° above the horizontal. Find the vertical and horizontal components of the velocity. 2. Pitcher Aroldis Chapman threw a pitch with a recorded velocity of 105 miles per hour. Assuming he threw the pitch at an angle of
Forces with magnitudes of 125 newtons and 300 newtons act on a hook (see figure). The angle between the two forces is 45°. Find the direction and magnitude of the resultant of these forces. (Hint: Write the vector representing each force in component form, then add the vectors.)
Forces with magnitudes of 2000newtons and 900 newtons act on a machine part at angles of 30° and −45°, respectively, with the positive x-axis (see figure). Find the direction and magnitude of the resultant of these forces.
Three forces with magnitudes of 75 pounds, 100 pounds, and 125 pounds act on an object at angles of 30°, 45°, and 120°, respectively, with the positive x-axis. Find the direction and magnitude of the resultant of these forces.
Three forces with magnitudes of 70pounds, 40 pounds, and 60 pounds act on an object at angles of −30°, 45°, and 135°, respectively, with the positive x-axis. Find the direction and magnitude of the resultant of these forces.
The cranes shown in the figure are lifting an object that weighs 20,240 pounds. Find the tension (in pounds) in the cable of each crane.
Repeat Exercise 83 for θ1 = 35.6° and θ2 = 40.4°.
A tetherball weighing 1 pound is pulled outward from the pole by a horizontal force u until the rope makes a 45° angle with the pole (see igure). Determine the resulting tension (in pounds) in the rope and the magnitude of u.
Use the figure to determine the tension (in pounds) in each cable supporting the load.

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