Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Precalculus
Artillery A projectile fired into the first quadrant from the origin of a coordinate system will pass through the point ( x,y ) at time t according to the relationship cot= 2x 2y+g t 2 where = the angle of elevation of the launcher and g= the acceleration due to gravity =32.2feet/secon d 2 . An artilleryman is firing at an enemy bunker located 2450 feet up the side of a hill that is 6175 feet away. He fires a round, and exactly 2.27 seconds later he scores a direct hit. (a) What angle of elevation did he use? (b) If the angle of elevation is also given by sec= v 0 t x , where v 0 is the muzzle velocity of the weapon, find the muzzle velocity of the artillery piece he used.82AYU83AYU84AYU85AYU86AYU1AYU2AYU3AYU4AYU5AYU6AYU7AYU8AYU9AYU10AYUIn Problems 13-36, solve each equation on the interval 02 . 2sin+3=2In Problems 13-36, solve each equation on the interval 02 . 1cos= 1 2In Problems 13-36, solve each equation on the interval 02 . 2sin+1=0In Problems 13-36, solve each equation on the interval 02 . cos+1=0In Problems 13-36, solve each equation on the interval 02 . tan+1=0In Problems 13-36, solve each equation on the interval 02 . 3 cot+1=0In Problems 13-36, solve each equation on the interval 02 . 4sec+6=2In Problems 13-36, solve each equation on the interval 02 . 5csc3=2In Problems 13-36, solve each equation on the interval 02 . 3 2 cos+2=1In Problems 13-36, solve each equation on the interval 02 . 4sin+3 3 = 3In Problems 13-36, solve each equation on the interval 02 . 4 cos 2 =1In Problems 13-36, solve each equation on the interval 02 . tan 2 = 1 3In Problems 13-36, solve each equation on the interval 02 . 2 sin 2 1=0In Problems 13-36, solve each equation on the interval 02 . 4 cos 2 3=0In Problems 13-36, solve each equation on the interval 02 . sin( 3 )=1In Problems 13-36, solve each equation on the interval 02 . tan 2 = 3In Problems 13-36, solve each equation on the interval 02 . cos( 2 )= 1 2In Problems 13-36, solve each equation on the interval 02 . tan( 2 )=1In Problems 13-36, solve each equation on the interval 02 . sec 3 2 =2In Problems 13-36, solve each equation on the interval 02 . cot 2 3 = 3In Problems 13-36, solve each equation on the interval 02 . cos( 2 2 )=132AYUIn Problems 13-36, solve each equation on the interval 02 . tan( 2 + 3 )=1In Problems 13-36, solve each equation on the interval 02 . cos( 3 4 )= 1 2In Problems 37-46, solve each equation. Give a general formula for all the solutions. List six solutions. sin= 1 2In Problems 37-46, solve each equation. Give a general formula for all the solutions. List six solutions. tan=1In Problems 37-46, solve each equation. Give a general formula for all the solutions. List six solutions. tan= 3 3In Problems 37-46, solve each equation. Give a general formula for all the solutions. List six solutions. cos= 3 2In Problems 37-46, solve each equation. Give a general formula for all the solutions. List six solutions. cos=0In Problems 37-46, solve each equation. Give a general formula for all the solutions. List six solutions. sin= 2 241AYU42AYU43AYU44AYUIn Problems 47-58, use a calculator to solve each equation on the interval 02 . Round answers to two decimal places. sin=0.4In Problems 47-58, use a calculator to solve each equation on the interval 02 . Round answers to two decimal places. cos=0.6In Problems 47-58, use a calculator to solve each equation on the interval 02 . Round answers to two decimal places. tan=5In Problems 47-58, use a calculator to solve each equation on the interval 02 . Round answers to two decimal places. cot=2In Problems 47-58, use a calculator to solve each equation on the interval 02 . Round answers to two decimal places. cos=0.9In Problems 47-58, use a calculator to solve each equation on the interval 02 . Round answers to two decimal places. sin=0.2In Problems 47-58, use a calculator to solve each equation on the interval 02 . Round answers to two decimal places. sec=4In Problems 47-58, use a calculator to solve each equation on the interval 02 . Round answers to two decimal places. csc=3In Problems 47-58, use a calculator to solve each equation on the interval 02 . Round answers to two decimal places. 5tan+9=054AYUIn Problems 47-58, use a calculator to solve each equation on the interval 02 . Round answers to two decimal places. 3sin2=0In Problems 47-58, use a calculator to solve each equation on the interval 02 . Round answers to two decimal places. 4cos+3=0In Problems 59-82, solve each equation on the interval 02 . 2 cos 2 +cos=0In Problems 59-82, solve each equation on the interval 02 . sin 2 1=0In Problems 59-82, solve each equation on the interval 02 . 2 sin 2 sin1=0In Problems 59-82, solve each equation on the interval 02 . 2 cos 2 +cos1=0In Problems 59-82, solve each equation on the interval 02 . ( tan1 )( sec1 )=0In Problems 59-82, solve each equation on the interval 02 . ( cot+1 )( csc 1 2 )=0In Problems 59-82, solve each equation on the interval 02 . sin 2 cos 2 =1+cosIn Problems 59-82, solve each equation on the interval 02 . cos 2 sin 2 +sin=0In Problems 59-82, solve each equation on the interval 02 . sin 2 =6( cos( )+1 )In Problems 59-82, solve each equation on the interval 02 . 2 sin 2 =3( 1cos( ) )In Problems 59-82, solve each equation on the interval 02 . cos=sin( )In Problems 59-82, solve each equation on the interval 02 . cossin( )=0In Problems 59-82, solve each equation on the interval 02 . tan=2sinIn Problems 59-82, solve each equation on the interval 02 . tan=cotIn Problems 59-82, solve each equation on the interval 02 . 1+sin=2 cos 2In Problems 59-82, solve each equation on the interval 02 . sin 2 =2cos+2In Problems 59-82, solve each equation on the interval 02 . 2 sin 2 5sin+3=0In Problems 59-82, solve each equation on the interval 02 . 2 cos 2 7cos4=0In Problems 59-82, solve each equation on the interval 02 . 3( 1cos )= sin 276AYU77AYU78AYUIn Problems 59-82, solve each equation on the interval 02 . sec 2 +tan=0In Problems 59-82, solve each equation on the interval 02 . sec=tan+cotIn Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. x+5cosx=0In Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. x4sinx=083AYU84AYU85AYU86AYUIn Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. x 2 2cosx=088AYUIn Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. x 2 2sin( 2x )=3x90AYU91AYUIn Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. 4cos( 3x ) e x =1 , x093AYU94AYU95AYU96AYU97AYU98AYU99AYU100AYU101AYU102AYUBlood Pressure Blood pressure is a way of measuring the amount of force exerted on the walls of blood vessels. It is measured using two numbers: systolic (as the heart beats) blood pressure and diastolic (as the heart rests) blood pressure. Blood pressures vary substantially from person to person, but a typical blood pressure is 120/80, which means the systolic blood pressure is 120 mmHg and the diastolic blood pressure is 80 mmHg. Assuming that a person’s heart beats 70 times per minute, the blood pressure P of an individual after t seconds can be modeled by the function P( t )=100+20sin( 7 3 t ) a. In the interval [ 0,1 ] , determine the times at which the blood pressure is 100 mmHg. b. In the interval [ 0,1 ] , determine the times at which the blood pressure is 120 mmHg. c. In the interval [ 0,1 ] , determine the times at which the blood pressure is between 100 and 105 mmHg.The Ferris Wheel In 1893, George Ferris engineered the Ferris wheel. It was 250 feet in diameter. If a Ferris wheel makes 1 revolution every 40 seconds, then the function h( t )=125sin( 0.157t 2 )+125 represents the height h , in feet, of a seat on the wheel as a function of time t , where t is measured in seconds. The ride begins when t=0 . a. During the first 40 seconds of the ride, at what time t is an individual on the Ferris wheel exactly 125 feet above the ground? b. During the first 80 seconds of the ride, at what time t is an individual on the Ferris wheel exactly 250 feet above the ground? c. During the first 40 seconds of the ride, over what interval of time t is an individual on the Ferris wheel more than 125 feet above the ground?Holding Pattern An airplane is asked to slay within a holding pattern near Chicago’s O'Hare International Airport. The function d( x )=70sin( 0.65x )+150 represents the distance d , in miles, of the airplane from the airport at lime x , in minutes. a. When the plane enters the holding pattern, x=0 , how far is it from Ο’Hare? b. During the first 20 minutes after the plane enters the holding pattern, at what time x is the plane exactly 100 miles from the airport? c. During the first 20 minutes after the plane enters the holding pattern, at what time x is the plane more than 100 miles from the airport? d. While the plane is in the holding pattern, will it ever be within 70 miles of the airport? Why?Projectile Motion A golfer hits a golf ball with an initial velocity of 100 miles per hour. The range R of the ball as a function of the angle to the horizontal is given by R( )=672sin( 2 ) , where R is measured in feet. a. At what angle should the ball be hit if the golfer wants the ball to travel 450 feet (150 yards)? b. At what angle should the ball be hit if the golfer wants the ball to travel 540 feet (180 yards)? c. At what angle should the ball be hit if the golfer wants the ball to travel at least 480 feet (160 yards)? d. Can the golfer hit the ball 720 feet (240 yards)?Heat Transfer In the study of heat transfer, the equation x+tanx=0 occurs. Graph Y 1 =x and Y 2 =tanx for x0 . Conclude that there are an infinite number of points of intersection of these two graphs. Now find the first two positive solutions of x+tanx=0 rounded to two decimal places.Carrying a Ladder around a Corner Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. See the illustration. It can be shown that the length L of the ladder as a function of is L( )=4csc+3sec . a. In calculus, you will be asked to find the length of the longest ladder that can turn the comer by solving the equation 3sectan4csccot=0 0 90 Solve this equation for . b. What is the length of the longest ladder that can be carried around the corner? c. Graph L=L( ) , 0 90 , and find the angle that minimizes the length L . d. Compare the result with the one found in part (a). Explain why the two answers are the same.Projectile Motion The horizontal distance that a projectile will travel in the air (ignoring air resistance) is given by the equation R( )= v 0 2 sin( 2 ) g where v 0 is the initial velocity of the projectile, is the angle of elevation, and g is acceleration due to gravity ( 9.8 meters per second squared). a. If you can throw a baseball with an initial speed of 34.8 meters per second, at what angle of elevation should you direct the throw so that the ball travels a distance of 107 meters before striking the ground? b. Determine the maximum distance that you can throw the ball. c. Graph R=R( ) , with v 0 =34.8 meters per second. d. Verify the results obtained in parts (a) and (b) using a graphing utility.110AYU111AYU112AYU113AYU114AYU115AYU116AYU117AYU118AYU119AYU120AYUTrue or False sin 2 =1 cos 2True or False sin( )+cos( )=cossinSuppose that fandg are two functions with the same domain. If f( x )=g( x ) for every x in the domain, the equation is called a( n ) _________. Otherwise, it is called a( n ) equation.tan 2 sec 2 = _____.cos()cos= _____.True or False sin( )+sin=0 for any value of .True or False In establishing an identity, it is often easiest to just multiply both sides by a well-chosen nonzero expression involving the variable.Which of the following equation is not an identity? (a) cot 2 +1= csc 2 (b) tan( )=tan (c) tan= cos sin (d) csc= 1 sinIn Problems 11-20, simplify each trigonometric expression by following the indicated direction. Rewrite in terms of sine and cosine functions: tancscIn Problems 11-20, simplify each trigonometric expression by following the indicated direction. Rewrite in terms of sine and cosine functions: cotsecIn Problems 11-20, simplify each trigonometric expression by following the indicated direction. Multiply cos 1sin by 1+sin 1+sin .In Problems 11-20, simplify each trigonometric expression by following the indicated direction. Multiply sin 1+cos by 1cos 1cos .In Problems 11-20, simplify each trigonometric expression by following the indicated direction. Rewrite over a common denominator: sin+cos cos + cossin sinIn Problems 11-20, simplify each trigonometric expression by following the indicated direction. Rewrite over a common denominator: 1 1cos + 1 1+cos .In Problems 11-20, simplify each trigonometric expression by following the indicated direction. Multiply and simplify: ( sin+cos )( sin+cos )1 sincosIn Problems 11-20, simplify each trigonometric expression by following the indicated direction. Multiply and simplify: ( tan+1 )( tan+1 ) sec 2 tanIn Problems 11-20, simplify each trigonometric expression by following the indicated direction. Factor and simplify: 3 sin 2 +4sin+1 sin 2 +2sin+1In Problems 11-20, simplify each trigonometric expression by following the indicated direction. Factor and simplify: cos 2 1 cos 2 cosestablish each identity. secsin=tanestablish each identity. secsin=tanestablish each identity. 1+ tan 2 ( )= sec 2establish each identity. 1+ cot 2 ( )= csc 2establish each identity. cos( tan+cot )=cscestablish each identity. sin( cot+tan )=secestablish each identity. tanucotu cos 2 u= sin 2 uestablish each identity. sinucscu cos 2 u= sin 2 uestablish each identity. ( sec1 )( sec+1 )= tan 2establish each identity. ( csc1 )( csc+1 )= cot 2establish each identity. ( sec+tan )( sectan )=1establish each identity. ( csc+cot )( csccot )=1establish each identity. cos 2 ( 1+ tan 2 )=1establish each identity. ( 1 cos 2 )( 1+ cot 2 )=1establish each identity. ( sin+cos ) 2 + ( sincos ) 2 =2establish each identity. tan 2 cos 2 + cot 2 sin 2 =1establish each identity. sec 4 sec 2 = tan 4 + tan 2establish each identity. csc 4 csc 2 = cot 4 + cot 2establish each identity. secutanu= cosu 1+sinuestablish each identity. cscucotu= sinu 1+cosuestablish each identity. 3 sin 2 +4 cos 2 =3+ cos 2establish each identity. 9 sec 2 5 tan 2 =5+4 sec 2establish each identity. 1 cos 2 1+sin =sinestablish each identity. 1 sin 2 1cos =cosestablish each identity. 1+tan 1tan = cot+1 cot1establish each identity. csc1 csc+1 = 1sin 1+sinestablish each identity. sec csc + sin cos =2tanestablish each identity. csc1 cot = cot csc+1establish each identity. 1+sin 1sin = csc+1 csc1establish each identity. cos+1 cos1 = 1+sec 1secestablish each identity. 1sin cos + cos 1sin =2secestablish each identity. cos 1+sin + 1+sin cos =2secestablish each identity. sin sincos = 1 1cotestablish each identity. 1 sin 2 1+cos =cosestablish each identity. 1sin 1+sin = ( sectan ) 254AYUestablish each identity. cos 1tan + sin 1cot =sin+cosestablish each identity. cot 1tan + tan 1cot =1+tan+cotestablish each identity. tan+ cos 1+sin =secestablish each identity. tan+ cos 1+sin =secestablish each identity. tan+sec1 tansec+1 =tan+secestablish each identity. sincos+1 sin+cos1 = sin+1 cosestablish each identity. tancot tan+cot = sin 2 cos 262AYUestablish each identity. tanucotu tanu+cotu +1=2 sin 2 u64AYU65AYU66AYUestablish each identity. 1 tan 2 1+ tan 2 +1=2 cos 2establish each identity. 1 cot 2 1+ cot 2 +2 cos 2 =1establish each identity. seccsc seccsc =sincosestablish each identity. sin 2 tan cos 2 cot = tan 2establish each identity. seccos=sintanestablish each identity. tan+cot=seccscestablish each identity. 1 1sin + 1 1+sin =2 sec 2establish each identity. 1+sin 1sin 1sin 1+sin =4tansecestablish each identity. sec 1sin = 1+sin cos 376AYU77AYUestablish each identity. sec 2 tan 2 +tan sec =sin+cosestablish each identity. sin+cos cos sincos sin =seccsc80AYU81AYUestablish each identity. sin 3 +co s 3 12 cos 2 = secsin tan1establish each identity. co s 2 sin 2 1 tan 2 = cos 284AYU85AYUestablish each identity. 12 cos 2 sincos =tancotestablish each identity. 1+sin+cos 1+sincos = 1+cos sin88AYU89AYUestablish each identity. ( 2asincos ) 2 + a 2 ( cos 2 sin 2 ) 2 = a 2establish each identity. tan+tan cot+cot =tantanestablish each identity. ( tan+tan )( 1cotcot )+( cot+cot )( 1tantan )=093AYU94AYUestablish each identity. ln| sec |=ln| cos |96AYUestablish each identity. ln| 1+cos |+ln| 1cos |=2ln| sin |98AYUIn Problems 101-104, show that the functions f and g are identically equal. f(x)=sinxtanxg(x)=secxcosx100AYU101AYU102AYU103AYU104AYU105AYU106AYU107AYU108AYUThe distance d from the point ( 2,3 ) to the point ( 5,1 ) is ____ . (pp. 4-6)If sin= 4 5 and is in quadrant II, then cos= ________. (pp. 401-403)(a) sin 4 cos 3 = _____ . (pp. 382-385) (b) tan 4 sin 6 = _____ . (pp. 382-385)If sin= 4 5 , 3 2 then cos= ____ . (pp.401-403)5AYU6AYU7AYU8AYU9AYU10AYU11AYU12AYU13AYU14AYU15AYU16AYU17AYU18AYU19AYU20AYU21AYU22AYUFind the exact value of each expression. sin 20 cos 10 +cos20 sin 10Find the exact value of each expression. sin 20 cos 80 cos20 sin 80Find the exact value of each expression. cos 70 cos 20 sin70 sin 20Find the exact value of each expression. cos 40 cos 10 +sin40 sin 10Find the exact value of each expression. tan 20 +tan25 1 tan20 tan25Find the exact value of each expression. tan 40 tan10 1 +tan40 tan10Find the exact value of each expression. sin 12 cos 7 12 cos 12 sin 7 12Find the exact value of each expression. cos 5 12 cos 7 12 sin 5 12 sin 7 12Find the exact value of each expression. cos 12 cos 5 12 +sin 5 12 sin 12Find the exact value of each expression. sin 18 cos 5 18 +cos 18 sin 5 18In Problems 35-40, find the exact value of each of the following under the given conditions: (a) sin( + ) (b) cos( + ) (c) sin( ) (d) tan( ) sin= 3 5 ,0 2 ;cos= 2 5 5 , 2 0In Problems 35-40, find the exact value of each of the following under the given conditions: (a) sin( + ) (b) cos( + ) (c) sin( ) (d) tan( ) cos= 5 5 ,0 2 ;sin= 4 5 , 2 0In Problems 35-40, find the exact value of each of the following under the given conditions: (a) sin( + ) (b) cos( + ) (c) sin( ) (d) tan( ) tan= 4 3 , 2 ;cos= 1 2 ,0 2In Problems 35-40, find the exact value of each of the following under the given conditions: (a) sin( + ) (b) cos( + ) (c) sin( ) (d) tan( ) tan= 5 12 , 3 2 ;sin= 1 2 , 3 2In Problems 35-40, find the exact value of each of the following under the given conditions: (a) sin( + ) (b) cos( + ) (c) sin( ) (d) tan( ) sin= 5 13 , 3 2 ;tan= 3 , 2In Problems 35-40, find the exact value of each of the following under the given conditions: (a) sin( + ) (b) cos( + ) (c) sin( ) (d) tan( ) cos= 1 2 , 2 0;sin= 1 3 ,0 2If sin= 1 3 , in quadrant II, find the exact value of: (a) cos (b) sin( + 6 ) (c) cos( 3 ) (d) tan( + 4 )If cos= 1 4 , in quadrant IV, find the exact value of: (a) sin (b) sin( 6 ) (c) cos( + 3 ) (d) tan( 4 )In problems 43-48, use the figures to evaluate each function if f( x )=sinx,g( x )=cosx,h( x )=tanx . f( + )In problems 43-48, use the figures to evaluate each function if f( x )=sinx,g( x )=cosx,h( x )=tanx . g( + )In problems 43-48, use the figures to evaluate each function if f( x )=sinx,g( x )=cosx,h( x )=tanx . g( )In problems 43-48, use the figures to evaluate each function if f( x )=sinx,g( x )=cosx,h( x )=tanx . f( )45AYUIn problems 43-48, use the figures to evaluate each function if f( x )=sinx,g( x )=cosx,h( x )=tanx . h( )establish each identify. sin( 2 + )=cosestablish each identify. cos( 2 + )=sin49AYU50AYUestablish each identify. sin( + )=sin52AYUestablish each identify. tan( )=tanestablish each identify. tan( 2 )=tan55AYU56AYU57AYUestablish each identify. cos( + )+cos( )=2coscosestablish each identify. sin( + ) sincos =1+cottanestablish each identify. sin( + ) coscos =tan+tanestablish each identify. cos( + ) coscos =1tantanestablish each identify. cos( ) sincos =cot+tanestablish each identify. sin( + ) sin( ) = tan+tan tantanestablish each identify. cos( + ) cos( ) = 1tantan 1+tantanestablish each identify. cot( + )= cotcot1 cot+cotestablish each identify. cot( )= cotcot+1 cotcot