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All Textbook Solutions for Precalculus Enhanced with Graphing Utilities
10CR11CR12CRWhat is the domain and the range of y=sinx ? (p. 395)A suitable restriction on the domain of the function f( x )= (x1) 2 to make it one-to-one would be_______. (pp.273-274)If the domain of a one-to-one function is [ 3, ) , the range of its inverse is________. (pp. 269-270)True or False The graph of y=cosx is decreasing on the interval [ 0, ] .(pp. 410-411)tan 4 = ______; sin 3 = ______(pp. 382-385)6AYUy= sin 1 x means _____, where 1x1 and 2 y 2 .cos 1 (cosx)=x , where__________.tan( tan 1 x )=x , where ______.True or FalseThe domain of y= sin 1 x is 2 x 2 .True or False sin( sin 1 0 )=0 and cos( cos 1 0 )=0 .True or False y= tan 1 x means x=tany , where x and 2 y 2 .In Problems 15-26, find, the exact value sin 1 0In Problems 15-26, find, the exact value of each expression. cos 1 1In Problems 15-26, find, the exact value of each expression. sin 1 ( 1 )In Problems 15-26, find, the exact value of each expression. cos 1 ( 1 )In Problems 15-26, find, the exact value of each expression. tan 1 0In Problems 15-26, find, the exact value of each expression. tan 1 ( 1 )In Problems 15-26, find, the exact value of each expression. sin 1 2 2In Problems 15-26, find, the exact value of each expression. tan 1 3 3In Problems 15-26, find, the exact value of each expression. tan 1 3In Problems 15-26, find, the exact value of each expression. sin 1 ( 3 2 )In Problems 15-26, find, the exact value of each expression. cos 1 ( 3 2 )In Problems 15-26, find, the exact value of each expression. sin 1 ( 2 2 )In Problems 27-38, use a calculator to find the value of each expression rounded to two decimal places. sin 1 0.1In Problems 27-38, use a calculator to find the value of each expression rounded to two decimal places. cos 1 0.6In Problems 27-38, use a calculator to find the value of each expression rounded to two decimal places. tan 1 5In Problems 27-38, use a calculator to find the value of each expression rounded to two decimal places. tan 1 0.2In Problems 27-38, use a calculator to find the value of each expression rounded to two decimal places. cos 1 7 8In Problems 27-38, use a calculator to find the value of each expression rounded to two decimal places. sin 1 1 8In Problems 27-38, use a calculator to find the value of each expression rounded to two decimal places. tan 1 ( 0.4 )In Problems 27-38, use a calculator to find the value of each expression rounded to two decimal places. tan 1 ( 3 )In Problems 27-38, use a calculator to find the value of each expression rounded to two decimal places. sin 1 ( 0.12 )In Problems 27-38, use a calculator to find the value of each expression rounded to two decimal places. cos 1 ( 0.44 )In Problems 27-38, use a calculator to find the value of each expression rounded to two decimal places. cos 1 2 3In Problems 27-38, use a calculator to find the value of each expression rounded to two decimal places. sin 1 3 5In Problems 39-62, find the exact value, if any, of each composite function. If there is no value, say it is not defined. Do not use a calculator. tan 1 [ tan( 3 8 ) ]In Problems 39-62, find the exact value, if any, of each composite function. If there is no value, say it is not defined. Do not use a calculator. cos 1 ( cos 4 5 )In Problems 39-62, find the exact value, if any, of each composite function. If there is no value, say it is not defined. Do not use a calculator. tan 1 [ tan( 3 8 ) ]In Problems 39-62, find the exact value, if any, of each composite function. If there is no value, say it is not defined. Do not use a calculator. sin 1 [ sin( 3 7 ) ]In Problems 39-62, find the exact value, if any, of each composite function. If there is no value, say it is not defined. Do not use a calculator. sin 1 ( sin 9 8 )In Problems 39-62, find the exact value, if any, of each composite function. If there is no value, say it is not defined. Do not use a calculator. cos 1 [ cos( 5 3 ) ]In Problems 39-62, find the exact value, if any, of each composite function. If there is no value, say it is not defined. Do not use a calculator. tan 1 ( tan 4 5 )In Problems 39-62, find the exact value, if any, of each composite function. If there is no value, say it is not defined. Do not use a calculator. tan 1 [ tan( 2 3 ) ]45AYU46AYU47AYU48AYU49AYU50AYU51AYU52AYUIn Problems 63-70, find the inverse function f 1 of each function f . Find the range of f and the domain and range of f 1 . f( x )=5sinx+2; 2 x 2In Problems 63-70, find the inverse function f 1 of each function f . Find the range of f and the domain and range of f 1 . f( x )=2tanx3; 2 x 2In Problems 63-70, find the inverse function f 1 of each function f . Find the range of f and the domain and range of f 1 . f( x )=2cos( 3x );0x 3In Problems 63-70, find the inverse function f 1 of each function f . Find the range of f and the domain and range of f 1 . f( x )=3sin( 2x ); 4 x 4In Problems 63-70, find the inverse function f 1 of each function f . Find the range of f and the domain and range of f 1 . f( x )=tan( x+1 )3;1 2 x 2 1In Problems 63-70, find the inverse function f 1 of each function f . Find the range of f and the domain and range of f 1 . f( x )=cos( x+2 )+1;2x2In Problems 63-70, find the inverse function f 1 of each function f . Find the range of f and the domain and range of f 1 . f( x )=cos( x+2 )+1;2x2In Problems 63-70, find the inverse function f 1 of each function f . Find the range of f and the domain and range of f 1 . f( x )=2cos( 3x+2 ); 2 3 x 2 3 + 3Find the exact solution of each equation. 4 sin 1 x=Find the exact solution of each equation. 2 cos 1 x=Find the exact solution of each equation. 3 cos 1 ( 2x )=2Find the exact solution of each equation. 6 sin 1 ( 3x )=Find the exact solution of each equation. 3 tan 1 x=In Problems 71-78, find the exact solution of each equation. 4 tan 1 x=In Problems 71-78, find the exact solution of each equation. 4 cos 1 x2=2 cos 1 xIn Problems 71-78, find the exact solution of each equation. 5 sin 1 x2=2 sin 1 x3In Problems 79-84, use the following discussion. The formula D=24[ 1 cos 1 ( tanitan ) ] can be used to approximate the number of hours of daylight D when the declination of the Sun is i at a location north latitude for any date between the vernal equinox and autumnal equinox. The declination of the Sun is defined as the angle i between the equatorial plane and any ray of light from the Sun. The latitude of a location is the angle between the Equator and the location on the surface of Earth, with the vertex of the angle located at the center of Earth. See the figure. To use the formula, cos 1 (tanitan) must be expressed in radians. Approximate the number of hours of daylight in Houston, Texas ( 29 45 north latitude), for the following dates: (a) Summer solstice ( i= 23.5 ) (b) Vernal equinox ( i= 0 ) (c) July 4 ( i= 22 48 )In Problems 79-84, use the following discussion. The formula D=24[ 1 cos 1 ( tanitan ) ] can be used to approximate the number of hours of daylight D when the declination of the Sun is i at a location north latitude for any date between the vernal equinox and autumnal equinox. The declination of the Sun is defined as the angle between the equatorial plane and ray of light from the Sun. The latitude of a location is the angle between the Equator and the location on the surface of Earth, with the vertex of the angle located at the center of Earth. See the figure. To use the formula, cos 1 ( tanitan ) must be expressed in radians. Approximate the number of hours of daylight in New York, New York ( 4045 north latitude), for the following dates: Summer solstice (i=23.5) Vernal equinox ( i=0 ) July 4 ( i=2248 )In Problems 79-84, use the following discussion. The formula D=24[ 1 cos 1 ( tanitan ) ] can be used to approximate the number of hours of daylight D when the declination of the Sun is i at a location north latitude for any date between the vernal equinox and autumnal equinox. The declination of the Sun is defined as the angle i between the equatorial plane and any ray of light from the Sun. The latitude of a location is the angle between the Equator and the location on the surface of Earth, with the vertex of the angle located at the center of Earth. See the figure. To use the formula, cos 1 (tanitan) must be expressed in radians. Approximate the number of hours of daylight in Honolulu, Hawaii ( 21 18 north latitude), for the following dates: (a) Summer solstice ( i= 23.5 ) (b) Vernal equinox ( i= 0 ) (c) July 4 ( i= 22 48 )In Problems 79-84, use the following discussion. The formula D=24[ 1 cos 1 ( tanitan ) ] can be used to approximate the number of hours of daylight D when the declination of the Sun is i at a location north latitude for any date between the vernal equinox and autumnal equinox. The declination of the Sun is defined as the angle i between the equatorial plane and any ray of light from the Sun. The latitude of a location is the angle between the Equator and the location on the surface of Earth, with the vertex of the angle located at the center of Earth. See the figure. To use the formula, cos 1 (tanitan) must be expressed in radians. Approximate the number of hours of daylight in Anchorage, Alaska ( 61 10 north latitude), for the following dates: (a) Summer solstice ( i= 23.5 ) (b) Vernal equinox ( i= 0 ) (c) July 4 ( i= 22 48 )In Problems 79-84, use the following discussion. The formula D=24[ 1 cos 1 ( tanitan ) ] can be used to approximate the number of hours of daylight D when the declination of the Sun is i at a location north latitude for any date between the vernal equinox and autumnal equinox. The declination of the Sun is defined as the angle i between the equatorial plane and any ray of light from the Sun. The latitude of a location is the angle between the Equator and the location on the surface of Earth, with the vertex of the angle located at the center of Earth. See the figure. To use the formula, cos 1 (tanitan) must be expressed in radians. Approximate the number of hours of daylight at the Equator ( 0 north latitude) for the following dates: Summer solstice ( i= 23.5 ) Vernal equinox i= 0 July 4 i= 0 ( i= 22 48 ) What do you conclude about the number of hours of daylight throughout the year for a location at the Equator?In Problems 79-84, use the following discussion. The formula D=24[ 1 cos 1 ( tanitan ) ] can be used to approximate the number of hours of daylight D when the declination of the Sun is i at a location north latitude for any date between the vernal equinox and autumnal equinox. The declination of the Sun is defined as the angle i between the equatorial plane and any ray of light from the Sun. The latitude of a location is the angle between the Equator and the location on the surface of Earth, with the vertex of the angle located at the center of Earth. See the figure. To use the formula, cos 1 (tanitan) must be expressed in radians. that is 66 30 north latitude for the following dates: (a) Summer solstice ( i= 23.5 ) (b) Vernal equinox ( i= 0 ) (c) July 4 ( i= 22 48 ) (d) Thanks to the symmetry of the orbital path of Earth around the Sun, the number of hours of daylight on the winter solstice may be found by computing the number of hours of daylight on the summer solstice and subtracting this result from 24 hours. Compute the number of hours of daylight for this location on the winter solstice. What do you conclude about daylight for a location at 66 30 north latitude?Being the First to See the Rising Sun Cadillac Mountain, elevation 1530 feet, is located in Acadia National Park. Maine, and is the highest peak on the east coast of the United States. It is said that a person standing on the summit will be the first person in the United States to see the rays of the rising Sun. How much sooner would a person atop Cadillac Mountain see the first rays than a person standing below, at sea level? [Hint: Consult the figure. When the person at D sees the first rays of the Sun, the person at F does not. The person at F sees the first rays of the Sun only after Earth has rotated so that F is at location Q . Compute the length of the arc subtended by the central angle . Then use the fact that at the latitude of Cadillac Mountain, in 24 hours a length of 2( 2710 )17,027.4 miles is subtended.]Movie Theater Screens Suppose that a movie theater has a screen that is 28 feet tall. When you sit down, the bottom of the screen is 6 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle. In the figure, is the viewing angle. Suppose that you sit x feet from the screen. The viewing angle is given by the function (x)= tan 1 ( 34 x ) tan 1 ( 6 x ) . What is your viewing angle if you sit 10 feet from the screen? 15 feel? 20 feel? If there are 5 feet between the screen and the first row of seats and there are 3 feet between each row and the row' behind it. which row results in the largest viewing angle? Using a graphing utility, graph ( x )= tan 1 ( 34 x ) tan 1 ( 6 x ) . What value of x results in the largest viewing angle?Area under a Curve The area under the graph of y= 1 1+ x 2 and above the x-axis between x=aandx=b is given by tan 1 b tan 1 a . See the figure. (a) Find the exact area under the graph of y= 1 1+ x 2 and above the x-axis between x=0andx= 3 . (b) Find the exact area under the graph of and above the x-axis between x= 3 3 andx=1 .Area under a Curve The area under the graph of y= 1 1 x 2 and above the x-axis between x=aandx=b is given by sin 1 b sin 1 a . See the figure. (a) Find the exact area under the graph of y= 1 1 x 2 and above the x-axis between x=0andx= 3 2 . (b) Find the exact area under the graph of y= 1 1 x 2 and above the x-axis between x= 1 2 andx= 1 2 .Problems 89 and 90 require the following discussion: The shortest distance between two points on Earth’s surface can be determined from the latitude and longitude of the two locations. For example, if location 1 has ( lat,lon )=( 1 , 1 ) and location 2 has ( lat,lon )=( 2 , 2 ) , the shortest distance between the two locations is approximately d=r cos 1 [ ( cos 1 cos 1 cos 2 cos 2 )+( cos 1 sin 1 cos 2 sin 2 )+( sin 1 sin 2 ) ] , where r=radiusofEarth3960 miles and the inverse cosine function is expressed in radians. Also, N latitude and E longitude are positive angles, and S latitude and W longitude are negative Source: www.infoplease.com Shortest Distance from Chicago to Honolulu Find the shortest distance from Chicago, latitude 41 50N , longitude 87 37W , to Honolulu, latitude 21 18N , longitude 157 50W . Round your answer to the nearest mile.Problems 89 and 90 require the following discussion: The shortest distance between two points on Earth's surface can be determined from the latitude and longitude of the two locations. For example, if location 1 has ( lat,lon )=( 1 , 1 ) and location 2 has ( lat,lon )=( 2 , 2 ) , the shortest distance between the two locations is approximately d=r cos 1 [ ( cos 1 cos 1 cos 2 cos 2 )+( cos 1 sin 1 cos 2 sin 2 )+( sin 1 sin 2 ) ] , where r=radiusofEarth3960 miles and the inverse cosine function is expressed in radians. Also, N latitude and E longitude are positive angles, and S latitude and W longitude are negative. Source: www. Infoplease.com Shortest Distance from Honolulu to Melbourne, Australia Find the shortest distance from Honolulu to Melbourne, Australia, latitude 3747S , longitude 14458E . Round your answer to the nearest mile.What is the domain and the range of y=secx ?True or False The graph of y=secx is one-to-one on the interval [ 0, 2 ) and on the interval ( 2 , ] . (pp. 427-428)If tan= 1 2 , 2 2 , then sin= ______.y= sec 1 x means ________, where | x | ______ and ______ y ______, y 2 .y= sec 1 x means ________, where | x | ______ and ______ y ______, y 2 .True or False It is impossible to obtain exact values for the inverse secant function.True or False csc 1 0.5 is not defined.True or False The domain of the inverse cotangent function is the set of real numbers.In Problems 9-36, find the exact value of each expression. cos( sin 1 2 2 )In Problems 9-36, find the exact value of each expression. sin( cos 1 1 2 )In Problems 9-36, find the exact value of each expression. tan[ cos 1 ( 3 2 ) ]In Problems 9-36, find the exact value of each expression. tan[ sin 1 ( 1 2 ) ]In Problems 9-36, find the exact value of each expression. sec( cos 1 1 2 )In Problems 9-36, find the exact value of each expression. cot[ sin 1 ( 1 2 ) ]In Problems 9-36, find the exact value of each expression. csc( tan 1 1 )In Problems 9-36, find the exact value of each expression. sec( tan 1 3 )In Problems 9-36, find the exact value of each expression. sin[ tan 1 ( 1 ) ]In Problems 9-36, find the exact value of each expression. cos[ sin 1 ( 3 2 ) ]In Problems 9-36, find the exact value of each expression. sec[ sin 1 ( 1 2 ) ]In Problems 9-36, find the exact value of each expression. csc[ cos 1 ( 3 2 ) ]In Problems 9-36, find the exact value of each expression. cos 1 ( sin 5 4 )In Problems 9-36, find the exact value of each expression. tan 1 ( cot 2 3 )In Problems 9-36, find the exact value of each expression. sin 1 [ cos( 7 6 ) ]In Problems 9-36, find the exact value of each expression. cos 1 [ tan( 4 ) ]In Problems 9-36, find the exact value of each expression. tan( sin 1 1 3 )In Problems 9-36, find the exact value of each expression. tan( cos 1 1 3 )In Problems 9-36, find the exact value of each expression. sec( tan 1 1 2 )In Problems 9-36, find the exact value of each expression. cos( sin 1 2 3 )In Problems 9-36, find the exact value of each expression. cot[ sin 1 ( 2 3 ) ]In Problems 9-36, find the exact value of each expression. csc[ tan 1 ( 2 ) ]In Problems 9-36, find the exact value of each expression. sin[ tan 1 ( 3 ) ]In Problems 9-36, find the exact value of each expression. cot[ cos 1 ( 3 3 ) ]In Problems 9-36, find the exact value of each expression. sec( sin 1 2 5 5 )In Problems 9-36, find the exact value of each expression. csc( tan 1 1 2 )In Problems 9-36, find the exact value of each expression. sin 1 ( cos 3 4 )In Problems 9-36, find the exact value of each expression. cos 1 ( sin 7 6 )In Problems 37-44, find the exact value of each expression. cot 1 3In Problems 37-44, find the exact value of each expression. cot 1 1In Problems 37-44, find the exact value of each expression. csc 1 ( 1 )In Problems 37-44, find the exact value of each expression. csc 1 2In Problems 37-44, find the exact value of each expression. sec 1 2 3 3In Problems 37-44, find the exact value of each expression. sec 1 ( 2 )In Problems 37-44, find the exact value of each expression. cot 1 ( 3 3 )In Problems 37-44, find the exact value of each expression. csc 1 ( 2 3 3 )In Problems 45-56, use a calculator to find the value of each expression rounded to two decimal places. sec 1 4In Problems 45-56, use a calculator to find the value of each expression rounded to two decimal places. csc 1 5In Problems 45-56, use a calculator to find the value of each expression rounded to two decimal places. cot 1 2In Problems 45-56, use a calculator to find the value of each expression rounded to two decimal places. sec 1 ( 3 )In Problems 45-56, use a calculator to find the value of each expression rounded to two decimal places. csc 1 ( 3 )In Problems 45-56, use a calculator to find the value of each expression rounded to two decimal places. cot 1 ( 1 2 )In Problems 45-56, use a calculator to find the value of each expression rounded to two decimal places. cot 1 ( 5 )52AYUIn Problems 45-56, use a calculator to find the value of each expression rounded to two decimal places. csc 1 ( 3 2 )In Problems 45-56, use a calculator to find the value of each expression rounded to two decimal places. sec 1 ( 4 3 )In Problems 45-56, use a calculator to find the value of each expression rounded to two decimal places. cot 1 ( 3 2 )In Problems 45-56, use a calculator to find the value of each expression rounded to two decimal places. cot 1 ( 10 )In Problems 57-66, write each trigonometric expression as an algebraic expression in u . cos( tan 1 u )In Problems 57-66, write each trigonometric expression as an algebraic expression in u . sin( cos 1 u )In Problems 57-66, write each trigonometric expression as an algebraic expression in u . tan( sin 1 u )In Problems 57-66, write each trigonometric expression as an algebraic expression in u . tan( cos 1 u )In Problems 57-66, write each trigonometric expression as an algebraic expression in u . sin( sec 1 u )In Problems 57-66, write each trigonometric expression as an algebraic expression in u . sin( cot 1 u )In Problems 57-66, write each trigonometric expression as an algebraic expression in u . cos( csc 1 u )In Problems 57-66, write each trigonometric expression as an algebraic expression in u . cos( sec 1 u )In Problems 57-66, write each trigonometric expression as an algebraic expression in u . tan( cot 1 u )In Problems 57-66, write each trigonometric expression as an algebraic expression in u . tan( sec 1 u )In Problems 67-78, f( x )=sinx , 2 x 2 , g( x )=cosx , 0x , and h( x )=tanx , 2 x 2 . Find the exact value of each composite function. g( f 1 ( 12 13 ) )In Problems 67-78, f( x )=sinx , 2 x 2 , g( x )=cosx , 0x , and h( x )=tanx , 2 x 2 . Find the exact value of each composite function. f( g 1 ( 5 13 ) )In Problems 67-78, f( x )=sinx , 2 x 2 , g( x )=cosx , 0x , and h( x )=tanx , 2 x 2 . Find the exact value of each composite function. g 1 ( f( 4 ) )In Problems 67-78, f( x )=sinx , 2 x 2 , g( x )=cosx , 0x , and h( x )=tanx , 2 x 2 . Find the exact value of each composite function. f 1 ( g( 5 6 ) )In Problems 67-78, f( x )=sinx , 2 x 2 , g( x )=cosx , 0x , and h( x )=tanx , 2 x 2 . Find the exact value of each composite function. h( f 1 ( 3 5 ) )In Problems 67-78, f( x )=sinx , 2 x 2 , g( x )=cosx , 0x , and h( x )=tanx , 2 x 2 . Find the exact value of each composite function. h( g 1 ( 4 5 ) )In Problems 67-78, f( x )=sinx , 2 x 2 , g( x )=cosx , 0x , and h( x )=tanx , 2 x 2 . Find the exact value of each composite function. g( h 1 ( 12 5 ) )In Problems 67-78, f( x )=sinx , 2 x 2 , g( x )=cosx , 0x , and h( x )=tanx , 2 x 2 . Find the exact value of each composite function. f( h 1 ( 5 12 ) )In Problems 67-78, f( x )=sinx , 2 x 2 , g( x )=cosx , 0x , and h( x )=tanx , 2 x 2 . Find the exact value of each composite function. g 1 ( f( 3 ) )In Problems 67-78, f( x )=sinx , 2 x 2 , g( x )=cosx , 0x , and h( x )=tanx , 2 x 2 . Find the exact value of each composite function. g 1 ( f( 6 ) )In Problems 67-78, f( x )=sinx , 2 x 2 , g( x )=cosx , 0x , and h( x )=tanx , 2 x 2 . Find the exact value of each composite function. h( g 1 ( 1 4 ) )In Problems 67-78, f( x )=sinx , 2 x 2 , g( x )=cosx , 0x , and h( x )=tanx , 2 x 2 . Find the exact value of each composite function. h( f 1 ( 2 5 ) )Problems 79 and 80 require the following discussion: When granular materials are allowed to fall freely, they form conical (cone-shaped) piles. The naturally occurring angle of slope, measured from the horizontal, at which the loose material comes to rest is called the angle of repose and varies for different materials. The angle of repose is related to the height h and base radius r of the conical pile by the equation = cot 1 r h . See the illustration. Angle of Repose: Deicing Salt Due to potential transportation issues (for example, frozen waterways) deicing salt used by highway departments in the Midwest must be ordered early and stored for future use. When deicing salt is stored in a pile 14 feet high, the diameter of the base of the pile is 45 feet. (a) Find the angle of repose for deicing salt. (b) What is the base diameter of a pile that is 17 feet high? (c) What is the height of a pile that has a base diameter of approximately 122 feet? Source: The Salt Storage Handbook, 2013Problems 79 and 80 require the following discussion: When granular materials are allowed to fall freely, they form conical (cone-shaped) piles. The naturally occurring angle of slope, measured from the horizontal, at which the loose material comes to rest is called the angle of repose and varies for different materials. The angle of repose is related to the height h and base radius r of the conical pile by the equation = cot 1 r h . See the illustration. Angle of Repose: Bunker Sand The steepness of sand bunkers on a golf course is affected by the angle of repose of the sand (a larger angle of repose allows for steeper bunkers). A freestanding pile of loose sand from a United States Golf Association (USGA) bunker had a height of 4 feet and a base diameter of approximately 6.68 feet. (a) Find the angle of repose for USGA bunker sand. (b) What is the height of such a pile if the diameter of the base is 8 feet? (c) A 6-foot-high pile of loose Tour Grade 50/50 sand has a base diameter of approximately 8.44 feet. Which type of sand (USGA or Tour Grade 50/50) would be better suited for steep bunkers?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.Using a graphing utility, graph y= cot 1 x .Using a graphing utility, graph y= sec 1 x .Using a graphing utility, graph y= csc 1 x .Explain in your own words how you would use your calculator to find the value of cot 1 10 .Consult three texts on calculus and write down the definition in each of y= sec 1 x and y= csc 1 x . Compare these with the definitions given in this text.Solve: 3x5=x+1sin( 4 )= ______; cos( 8 3 )= ______.Find the real solutions of 4 x 2 x5=0 .Find the real solutions of x 2 x1=0 .Find the real solutions of ( 2x1 ) 2 3( 2x1 )4=0 .6AYUTrue or False Most trigonometric equations have unique solutions.True or False Two solutions of the equation sin= 1 2 are 6 and 5 6 .True or False The set of all solutions of the equation tan=1 is given by { |= 4 +k,kisanyinteger }True or False The equation sin=2 has a real solution that can be found using a calculator.In 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 )=1In Problems 13-36, solve each equation on the interval 02 . sin( 3+ 18 )=1In 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 2In Problems 37-46, solve each equation. Give a general formula for all the solutions. List six solutions. cos( 2 )= 1 2In Problems 37-46, solve each equation. Give a general formula for all the solutions. List six solutions. sin( 2 )=1In Problems 37-46, solve each equation. Give a general formula for all the solutions. List six solutions. sin 2 = 3 2In Problems 37-46, solve each equation. Give a general formula for all the solutions. List six solutions. tan 2 =1In 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=0In Problems 47-58, use a calculator to solve each equation on the interval 02 . Round answers to two decimal places. 4cot=5In 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 2In Problems 59-82, solve each equation on the interval 02 . 4( 1+sin )= cos 2In Problems 59-82, solve each equation on the interval 02 . tan 2 = 3 2 secIn Problems 59-82, solve each equation on the interval 02 . csc 2 =cot+1In 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=0In Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. 22x17sinx=3In Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. 19x+8cosx=2In Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. sinx+cosx=xIn Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. sinxcosx=xIn Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. x 2 2cosx=0In Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. x 2 +3sinx=0In Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. x 2 2sin( 2x )=3xIn Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. x 2 =x+3cos( 2x )In Problems 83-94, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places. 6sinx e x =2 , x0In 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 , x0What are the zeros of f( x )=4 sin 2 x3 on the interval [ 0,2 ] ?What are the zeros of f( x )=2cos( 3x )+1 on the interval [ 0, ] ?f(x)=3sinx a. Find the zeros of f on the interval [ 2,4 ] . b. Graph f(x)=3sinx on the interval [ 2,4 ] . c. Solve f(x)= 3 2 on the interval [ 2,4 ] . What points are on the graph of f ? Label these points on the graph drawn in part (b). d. Use the graph drawn in part (b) along with the results of part (c) to determine the values of .v such that f(x) 3 2 on the interval [ 2,4 ] .f( x )=2cosx a. Find the zeros of f on the interval [ 2,4 ] . b. Graph f( x )=2cosx on the interval [ 2,4 ] . c. Solve f( x )= 3 on the interval [ 2,4 ] . What points are on the graph of f ? Label these points on the graph drawn in part (b). d. Use the graph drawn in part (b) along with the results of part (c) to determine the values of x such that f( x ) 3 on the interval [ 2,4 ] .f( x )=4tanx a. Solve f( x )=4 . b. For what values of x is f( x )4 on the interval ( 2 , 2 ) ?f( x )=cotx a. Solve f( x )= 3 . b. For what values of x is f( x ) 3 on the interval ( 0, ) ?a. Graph f( x )=3sin( 2x )+2 and g( x )= 7 2 on the same Cartesian plane for the interval [ 0, ] . b. Solve f( x )=g( x ) on the interval [ 0, ] , and label the points of intersection on the graph drawn in part (a). c. Solve f( x )g( x ) on the interval [ 0, ] . d. Shade the region bounded by f( x )=3sin( 2x )+2 and g( x )= 7 2 between the two points found in part (b) on the graph drawn in part (a).a. Graph f( x )=2cos x 2 +3 and g( x )=4 on the same Cartesian plane for the interval [ 0,4 ] . b. Solve f( x )=g( x ) on the interval [ 0,4 ] , and label the points of intersection on the graph drawn in part (a). c. Solve f( x )g( x ) on the interval [ 0,4 ] . d. Shade the region bounded by f( x )=2cos x 2 +3 and g( x )=4 between the two points found in part (b) on the graph drawn in part (a).a. Graph f( x )=4cosx and g( x )=2cosx+3 on the same Cartesian plane for the interval [ 0,2 ] . b. Solve f( x )=g( x ) on the interval [ 0,2 ] , and label the points of intersection on the graph drawn in part (a). c. Solve f( x )g( x ) on the interval [ 0,2 ] . d. Shade the region bounded by f( x )=4cosx and g( x )=2cosx+3 between the two points found in part (b) on the graph drawn in part (a).a. Graph f( x )=2sinx and g( x )=2sinx+2 on the same Cartesian plane for the interval [ 0,2 ] . b. Solve f( x )=g( x ) on the interval [ 0,2 ] , and label the points of intersection on the graph drawn in part (a). c. Solve f( x )g( x ) on the interval [ 0,2 ] . d. Shade the region bounded by f( x )=2sinx and g( x )=2sinx+2 between the two points found in part (b) on the graph drawn in part (a).Blood 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.Projectile Motion Refer to Problem 111. a. If you can throw a baseball with an initial speed of 40 meters per second, at what angle of elevation should you direct the throw so that the ball travels a distance of 110 meters before striking the ground? b. Determine the maximum distance that you can throw the ball. c. Graph R=R( ) , with v 0 =40 meters per second. d. Verify the results obtained in parts (a) and (b) using a graphing utility.111AYU112AYU113AYU114AYU115AYU116AYU117AYU118AYU119AYU120AYUTrue or False sin 2 =1 cos 22AYUSuppose 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 .