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All Textbook Solutions for Precalculus
establish each identify. sec( + )= csccsc cotcot1establish each identify. sec( )= secsec 1+tantanestablish each identify. sin( )sin( + )= sin 2 sin 2establish each identify. cos( )cos( + )= cos 2 sin 2establish each identify. sin( +k )= ( 1 ) k sin,k any integerestablish each identify. cos( +k )= ( 1 ) k cos,k any integerIn problems 75-86, find the exact value of each expression. sin( sin 1 1 2 + cos 1 0 )In problems 75-86, find the exact value of each expression. sin( sin 1 3 2 + cos 1 1 )In problems 75-86, find the exact value of each expression. sin[ sin 1 3 5 cos 1 ( 4 5 ) ]In problems 75-86, find the exact value of each expression. sin[ sin 1 ( 4 5 ) tan 1 3 4 ]In problems 75-86, find the exact value of each expression. cos( ta n 1 4 3 +cos 1 5 13 )In problems 75-86, find the exact value of each expression. cos[ tan 1 5 12 sin 1 ( 3 5 ) ]In problems 75-86, find the exact value of each expression. cos( sin 1 5 13 tan 1 3 4 )In problems 75-86, find the exact value of each expression. cos( tan 1 4 3 +cos 1 12 13 )In problems 75-86, find the exact value of each expression. tan( sin 1 3 5 + 6 )In problems 75-86, find the exact value of each expression. tan( 4 cos 1 3 5 )In problems 75-86, find the exact value of each expression. tan( sin 1 4 5 + cos 1 1 )In problems 75-86, find the exact value of each expression. tan( cos 1 4 5 +sin 1 1 )In Problems 87-92, write each trigonometric expression as an algebraic expression containing UandV . Give the restrictions required on UandV . cos( cos 1 u+ sin 1 v )In Problems 87-92, write each trigonometric expression as an algebraic expression containing UandV . Give the restrictions required on UandV . sin( sin 1 u cos 1 v )In Problems 87-92, write each trigonometric expression as an algebraic expression containing UandV . Give the restrictions required on UandV . sin( tan 1 u sin 1 v )In Problems 87-92, write each trigonometric expression as an algebraic expression containing UandV . Give the restrictions required on UandV . cos( tan 1 u +tan 1 v )In Problems 87-92, write each trigonometric expression as an algebraic expression containing UandV . Give the restrictions required on UandV . tan( sin 1 u cos 1 v )In Problems 87-92, write each trigonometric expression as an algebraic expression containing UandV . Give the restrictions required on UandV . sec( tan 1 u +cos 1 v )In problems 93-98, solve each equation on the interval 02 . sin 3 cos=1In problems 93-98, solve each equation on the interval 02 . 3 sin+cos=1In problems 93-98, solve each equation on the interval 02 . sin+cos= 2In problems 93-98, solve each equation on the interval 02 . sincos= 2In problems 93-98, solve each equation on the interval 02 . tan+ 3 =secIn problems 93-98, solve each equation on the interval 02 . cot+csc= 397AYU98AYU99AYU100AYU101AYU102AYU103AYU104AYU105AYU106AYU107AYU108AYU109AYU110AYU111AYUcos( 2 )= cos 2 =1=12.
tan 2 = 1cosTrue or False tan( 20 )= 2tan 1 tan 2True or False sin( 2 ) has two equivalent forms: 2sincos and si n 2 co s 2True or False tan( 2 )+tan( 2 )=tan( 4 )In Problems use the information given about the angle to find the exact value of:
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In Problems 920, use the information given about the angle ,02, to find the exact value of: (a)sin(2)(b)cos(2)(c)sin2(d)cos2 cos=35,02In Problems use the information given about the angle to find the exact value of:
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In Problems 920, use the information given about the angle ,02, to find the exact value of: (a)sin(2)(b)cos(2)(c)sin2(d)cos2 tan=12,32In Problems 920, use the information given about the angle ,02, to find the exact value of: (a)sin(2)(b)cos(2)(c)sin2(d)cos2 cos=63,2In Problems 920, use the information given about the angle ,02, to find the exact value of: (a)sin(2)(b)cos(2)(c)sin2(d)cos2 sin=33,322In Problems use the information given about the angle to find the exact value of:
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In Problems use the information given about the angle to find the exact value of:
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In Problems use the information given about the angle to find the exact value of:
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In Problems 920, use the information given about the angle ,02, to find the exact value of: (a)sin(2)(b)cos(2)(c)sin2(d)cos2 sec=2,csc0In Problems 920, use the information given about the angle ,02, to find the exact value of: (a)sin(2)(b)cos(2)(c)sin2(d)cos2 tan=3,sin0In Problems use the information given about the angle to find the exact value of:
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In Problems 21-30, use the Half-angle Formulas to find the exact value of each expression. sin 22.5In Problems 21-30, use the Half-angle Formulas to find the exact value of each expression. cos 22.5In Problems 21-30, use the Half-angle Formulas to find the exact value of each expression. tan 7 8In Problems 21-30, use the Half-angle Formulas to find the exact value of each expression. tan 9 8In Problems 21-30, use the Half-angle Formulas to find the exact value of each expression. cos 165In Problems 21-30, use the Half-angle Formulas to find the exact value of each expression. sin 195In Problems 21-30, use the Half-angle Formulas to find the exact value of each expression. sec 15 8In Problems 21-30, use the Half-angle Formulas to find the exact value of each expression. csc 7 8In Problems 21-30, use the Half-angle Formulas to find the exact value of each expression. sin( 8 )In Problems 21-30, use the Half-angle Formulas to find the exact value of each expression. cos( 3 8 )In Problems 31-42, use the figures to evaluate each function, given that f( x )=sinx , g( x )=cosx , and h( x )=tanx f( 2 )In Problems 31-42, use the figures to evaluate each function, given that f( x )=sinx , g( x )=cosx , and h( x )=tanx g( 2 )In Problems 31-42, use the figures to evaluate each function, given that f( x )=sinx , g( x )=cosx , and h( x )=tanx g( 2 )In Problems 31-42, use the figures to evaluate each function, given that f( x )=sinx , g( x )=cosx , and h( x )=tanx f( 2 )In Problems 31-42, use the figures to evaluate each function, given that f( x )=sinx , g( x )=cosx , and h( x )=tanx h( 2 )In Problems 31-42, use the figures to evaluate each function, given that f( x )=sinx , g( x )=cosx , and h( x )=tanx h( 2 )In Problems 31-42, use the figures to evaluate each function, given that f( x )=sinx , g( x )=cosx , and h( x )=tanx g( 2 )In Problems 31-42, use the figures to evaluate each function, given that f( x )=sinx , g( x )=cosx , and h( x )=tanx f( 2 )In Problems 31-42, use the figures to evaluate each function, given that f( x )=sinx , g( x )=cosx , and h( x )=tanx f( 2 )In Problems 31-42, use the figures to evaluate each function, given that f( x )=sinx , g( x )=cosx , and h( x )=tanx g( 2 )In Problems 31-42, use the figures to evaluate each function, given that f( x )=sinx , g( x )=cosx , and h( x )=tanx h( 2 )In Problems 31-42, use the figures to evaluate each function, given that f( x )=sinx , g( x )=cosx , and h( x )=tanx h( 2 )Show that sin 4 = 3 8 1 2 cos( 2 )+ 1 8 cos( 4 )Show that sin( 4 )=( cos )( 4sin8 sin 3 ) .43AYU44AYU45AYU46AYUcos 4 sin 4 =cos( 2 )48AYUestablish each identify. cot( 2 )= cot 2 -1 2cotestablish each identify. cot( 2 )= 1 2 ( cot-tan )establish each identify. sec( 2 )= sec 2 2- sec 252AYUestablish each identify. cos 2 ( 2u ) -sin 2 ( 2u )=cos( 4u )54AYUestablish each identify. cos( 2 ) 1+sin( 2 ) = cot-1 cot+1In Problemsestablish each identity.
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57AYU58AYUestablish each identify. cot 2 v 2 = secv+1 secv-160AYU61AYUestablish each identify. 1- 1 2 sin( 2 )= sin 3 +cos 3 sin+cos63AYU64AYUestablish each identify. tan( 3 )= 3tan tan 3 13 tan 266AYU67AYU68AYUsolve each equation on the interval 02 . cos( 2 )+6 sin 2 =4solve each equation on the interval 02 . cos( 2 )=22 sin 2solve each equation on the interval 02 . cos( 2 )=cossolve each equation on the interval 02 . sin( 2 )=cos73AYUsolve each equation on the interval 02 . cos( 2 )+cos( 4 )=075AYUsolve each equation on the interval 02 . cos( 2 )+5cos+3=077AYUsolve each equation on the interval 02 . tan( 2 )+2cos=0find the exact value of each expression. sin( 2 sin 1 1 2 )find the exact value of each expression. sin[ 2 sin 1 3 2 ]find the exact value of each expression. cos( 2 sin 1 3 5 )find the exact value of each expression. cos( 2 cos 1 4 5 )find the exact value of each expression. tan[ 2 cos 1 ( 3 5 ) ]find the exact value of each expression. tan( 2 tan 1 3 4 )85AYUfind the exact value of each expression. cos[ 2 tan 1 ( 4 3 ) ]87AYUfind the exact value of each expression. cos 2 ( 1 2 sin 1 3 5 )89AYUfind the exact value of each expression. csc[ 2 sin 1 ( 3 5 ) ]91AYU92AYU93AYUConstructing a Rain Gutter A rain gutter is to be constructed of aluminum sheets 12 inches wide. After marking off a length of 4 inches from each edge, the builder bends this length up at an angle . See the illustration. The area A of the opening as a function of is given by A( )=16sin( cos+1 ) , 0 90 (a) In calculus, you will be asked to find the angle that maximizes A by solving the equation cos( 2 )+cos =0,0 90 Solve this equation for . (b) What is the maximum area A of the opening? (c) Graph A=A( ), 0 90 and find the angle that maximizes the area A . Also find the maximum area.Laser Projection In a laser projection system, the optical angle or scanning angle is related to the throve distance D from the scanner to the screen and the projected image width W by the equation D= 1 2 w csccot (a) Show that the projected image width is given by W=2Dtan 2 (b) Find the optical angle if the throw distance is 15 feet and the projected image width is 6.5 feet. Source: Pangolin Laser Systems, Inc.96AYUProjectile Motion An object is propelled upward at an angle , 45 90 , to the horizontal with an initial velocity v 0 feet per second from the base of a plane that makes an angle of 45 with the horizontal. See the illustration. If air resistance is ignored, the distance R that it travels up the inclined plane is given by the function R( )= v 0 2 2 16 cos( sincos ) Show that R( )= v 0 2 2 32 [sin( 2 )cos( 2 )1] In calculus, you will be asked to find the angle that maximizes R by solving the equation sin( 2 )+cos( 2 )=0 solve the equation for . What is the maximum distance R if v 0 =32 feet per second? Graph R=R( ), 45 90 , and find the angle that maximizes the distance R . Also find the maximum distance. Use v 0 =32 feet per second. Compare the results with the answers found in parts (b) and (c).98AYU99AYUGeometry A rectangle is inscribed in a semicircle of radius 1 See the illustration. (a) Express the area A of the rectangle as a function of the angle shown in the illustration. (b) Show that A( )=sin( 2 ) . (c) Find the angle that results in the largest area A . (d) Find the dimensions of this largest rectangle.101AYU102AYU103AYU104AYUIf z=tan 2 , show that sin= 2z 1+ z 2 .106AYU107AYU108AYU109AYU110AYU111AYU112AYUfind the exact value of each expression. sin 195 cos75find the exact value of each expression. cos 285 cos195find the exact value of each expression. sin 195 cos75find the exact value of each expression. sin 75 +sin15Find the exact value of each expression. cos 225 cos 195Find the exact value of each expression. sin 255 sin 15express each product as a sum containing only sines or only cosines. sin( 4 )sin( 2 )express each product as a sum containing only sines or only cosines. cos( 4 )cos( 2 )express each product as a sum containing only sines or only cosines. sin( 4 )cos( 2 )express each product as a sum containing only sines or only cosines. sin( 3 )sin( 5 )express each product as a sum containing only sines or only cosines. cos( 3 )cos( 5 )express each product as a sum containing only sines or only cosines. sin( 4 )cos( 6 )express each product as a sum containing only sines or only cosines. sinsin( 2 )express each product as a sum containing only sines or only cosines. cos( 3 )cos( 4 )express each product as a sum containing only sines or only cosines. sin 3 2 cos 2express each product as a sum containing only sines or only cosines. sin 2 cos 5 2express each sum or difference as a product of sines and/or cosines. sin( 4 )-sin( 2 )express each sum or difference as a product of sines and/or cosines. sin( 4 )+sin( 2 )express each sum or difference as a product of sines and/or cosines. cos( 2 )+cos( 4 )20AYUexpress each sum or difference as a product of sines and/or cosines. sin+sin( 3 )22AYU23AYUexpress each sum or difference as a product of sines and/or cosines. sin 2 -sin 3 2establish each identify. sin+sin(3) 2sin(2) =cosestablish each identify. cos+cos(3) 2cos(2) =cosestablish each identify. sin(4)+sin(2) cos(4)+cos(2) =tan(3)28AYUestablish each identify. cos-cos(3) sin+sin(3) =tan30AYU31AYU32AYU33AYU34AYU35AYU36AYU37AYU38AYU39AYU40AYU41AYUestablish each identify. 1-cos( 2 )+cos( 4 )-cos( 6 )=4sincos( 2 )sin( 3 )43AYUsolve each equation on the interval 02 cos( 2 )+cos( 4 )=045AYUsolve each equation on the interval 02 sin( 4 )-sin( 6 )=047AYU48AYU49AYU50AYU51AYU52AYU53AYU54AYU55AYU1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE63RE64RE65RE66RE67RE68RE69RE70RE71RE72RE73RE74RE1CT2CTIn Problem, use the given information to determine the three remaining parts of each triangle.
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4CTIn Problem 35, use the given information to determine the three remaining parts of each triangle.6CT7CT8CT9CT10. Find the area of the triangle described in Problem 5.
11CTA hot- air balloon is flying at a height of 600 feet and is directly above the Marshall Space Flight Center in Huntsville, Alabama. The pilot of the balloon looks down at the airport that is known to be 5 miles from the Marshall Space Flight Center. What is the angle of depression from the balloon to the airport?Find the area of the shaded region enclosed in a semicircle of diameter 8 centimetres. The length of the chord AB is 6 centimetres. [Hint: Triangle ABC is a right triangle.]14. Find the area of the quadrilateral shown.