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All Textbook Solutions for Precalculus: Mathematics for Calculus - 6th Edition
32E33E34E35E36E37E38ESurveying To find the distance across a small lake, a surveyor has taken the measurements shown. Find the distance across the lake using this information.Geometry A parallelogram has sides of lengths 3 and 5, and one angle is 50. Find the lengths of the diagonals.Calculating Distance Two straight roads diverge at an angle of 65. Two cars leave the intersection at 2:00 p.m., one traveling at 50 mi/h and the other at 30 mi/h. How far apart are the cars at 2:30 p.m.?Calculating Distance A car travels along a straight road, heading east for 1 h, then traveling for 30 min on another road that leads northeast. If the car has maintained a constant speed of 40 mi/h, how far is it from its starting position?Dead Reckoning A pilot flies in a straight path for 1 h 30 min. She then makes a course correction, heading 10 to the right of her original course, and flies 2 h in the new direction. If she maintains a constant speed of 625 mi/h, how far is she from her starting position?Navigation Two boats leave the same port at the same time. One travels at a speed of 30 mi/h in the direction N 50E, and the other travels at a speed of 26 mi/h in a direction S 70 E (see the figure). How far apart are the two boats after 1 h?45ENavigation Airport B is 300 mi from airport A at a bearing N 50 E (see the figure). A pilot wishing to fly from A to B mistakenly flies due east at 200 mi/h for 30 min, when he notices his error. (a) How far is the pilot from his destination at the time he notices the error? (b) What bearing should he head his plane to arrive at airport B?Triangular Field A triangular field has sides of lengths 22, 36, and 44 yd. Find the largest angle.Towing a Barge Two tugboats that are 120 ft apart pull a barge, as shown. If the length of one cable is 212 ft and the length of the other is 230 ft, find the angle formed by the two cables.Flying Kites A boy is flying two kites at the same time. He has 380 ft of line out to one kite and 420 ft to the other. He estimates the angle between the two lines to be 30. Approximate the distance between the kites.Securing a Tower A 125-ft tower is located on the side of a mountain that is inclined 32 to the horizontal. A guy wire is to be attached to the top of the tower and anchored at a point 55 ft downhill from the base of the tower. Find the shortest length of wire needed.Cable Car A steep mountain is inclined 74 to the horizontal and rises 3400 ft above the surrounding plain. A cable car is to be installed from a point 800 ft from the base to the top of the mountain, as shown. Find the shortest length of cable needed.CN Tower The CN Tower in Toronto, Canada, is the tallest free-standing structure in North America. A woman on the observation deck, 1150 ft above the ground, wants to determine the distance between two landmarks on the ground below. She observes that the angle formed by the lines of sight to these two landmarks is 43. She also observes that the angle between the vertical and the line of sight to one of the landmarks is 62 and that to the other landmark is 54. Find the distance between the two landmarks.Land Value Land in downtown Columbia is valued at 20 a square foot. What is the value of a triangular lot with sides of lengths 112, 148, and 190 ft?54E1RCC2RCC3RCC4RCC5RCC6RCC7RCC8RCC1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE63RE64RE65RE66RE67RE68RE69RE70RE71RE72RE1T2TFind the exact value of each expression. (a) sin 8 cos 22 + cos 8 sin 22 (b) sin 75 (c) sin12For the angles and in the figures, find cos( + ).5T6T7T8TFind the exact value of cos(2tan1940).Rewrite the expression as an algebraic function of x and y: sin(cos1 x tan1 y).Wave on a Canal A wave on the surface of a long canal is described by the function y(x,t)=5sin(2x2t)x0 (a) Find the function that models the position of the point x = 0 at any time t. (b) Sketch the shape of the wave when t = 0, 0.4, 0.8, 1.2, and 1.6. Is this a traveling wave? (c) Find the velocity of the wave.2PTraveling Wave A traveling wave is graphed at the instant t = 0. If it is moving to the right with velocity 6, find an equation of the form y(x, t) = A sin(kx kvt) for this wave.Traveling Wave A traveling wave has period 2/3, amplitude 5, and velocity 0.5. (a) Find the equation of the wave. (b) Sketch the graph for t = 0, 0.5, 1, 1.5, and 2.Standing Wave A standing wave with amplitude 0.6 is graphed at several times t as shown in the figure. If the vibration has a frequency of 20 Hz, find an equation of the form y(x, t) = A sin x cos t that models this wave.6P7P8P1CRT2CRT3CRT4CRT5CRT6CRT7CRT8CRT9CRT10CRT11CRT12CRTAn 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 _______________.3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73E74E75E76E77E78E79E80E81E82E83E84E85E86E87E88E89E90E91E92E93E94E95E96E97E98E99E100E101E102E103E104EIf 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) = __________.3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63E