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All Textbook Solutions for Precalculus
For Exercises 43-58, graph the function. (See Example 3-4) y=cot2x+3For Exercises 43-58, graph the function. (See Example 3-4) y=tan3x4For Exercises 43-58, graph the function. (See Example 3-4) y=tan3x+456PE57PEFor Exercises 43-58, graph the function. (See Example 3-4) y=2cot2x2+1Write a function of the form y=tanBxC for the given graph.Write a function of the form y=cotBx for the given graph.For Exercises 61-64, given y=fx and y=gx . a. Find fgx and graph the resulting function. b. Find gfx and graph the resulting function. fx=tanx and gx=462PE63PEFor Exercises 61-64, given y=fx and y=gx . a. Find fgx and graph the resulting function. b. Find gfx and graph the resulting function. fx=x3 and gx=secx65PEFor Exercises 65-68, complete the statements for the function provided. fx=cotx a. As x0 ,fx b. As x0+ ,fxFor Exercises 65-68, complete the statements for the function provided. fx=cscx a. As x0 ,fx b. As x0+ ,fxFor Exercises 65-68, complete the statements for the function provided. fx=secx a. As x2 ,fx b. As x+2 ,fxExplain how to find two consecutive vertical asymptotes of y=AtanBxC forB0 .70PE71PE72PEFor Exercises 73-76, solve each equation for x on the interval 0,2. tanx=174PE75PEFor Exercises 73-76, solve each equation for x on the interval0,2. cotx=1Show that the maximum length L (in feet) of a beam that can fit around the corner shown in the figure is L=5sec+4csc .Graph the functions y=tanx forx and y=xx33+2x515 on the interval 2,2 . How do the functions compare for values of x taken close to 0 ?Graph the functions y=secx and y=1+x22+5x424 on the interval , . How do the functions compare for values of x taken close to 0 ?Given fx=x2,gx=tanx, and hx=secx . a. Find fhx . b. Graph gx and fhx together using the ZTRIG window. The relationship between the two graphs will be studied in calculus. For a given value of x in the domain of gx=tanx=tanx,y=sec2x gives the slope of a line tangent to g at x .Given rx=x2,sx=cotx, and tx=cscx , a. Find rtx b. Graph sx and rtx together on the ZTRIG window. The relationship between the two graphs will be studied in calculus. For a given value of x in the domain of sx=cotx,y=csc2x gives the slope of a line tangent to s at x .For Exercises 1-16, identify which functions shown here (f,g,h, and so on) have the given characteristics fx=sin2x+3gx=3cos12x4hx=3sin12x5kx=3sec2x+mx=2csc2x23nx=3tanx2px=2cot12x+tx=3+2cosx Has an amplitude of 32PREFor Exercises 1-16, identify which functions shown here (f,g,h, and so on) have the given characteristics fx=sin2x+3gx=3cos12x4hx=3sin12x5kx=3sec2x+mx=2csc2x23nx=3tanx2px=2cot12x+tx=3+2cosx Has no amplitudeFor Exercises 1-16, identify which functions shown here (f,g,h, and so on) have the given characteristics fx=sin2x+3gx=3cos12x4hx=3sin12x5kx=3sec2x+mx=2csc2x23nx=3tanx2px=2cot12x+tx=3+2cosx Has a period of 2For Exercises 1-16, identify which functions shown here (f,g,h, and so on) have the given characteristics fx=sin2x+3gx=3cos12x4hx=3sin12x5kx=3sec2x+mx=2csc2x23nx=3tanx2px=2cot12x+tx=3+2cosx Has a period of 46PREFor Exercises 1-16, identify which functions shown here (f,g,h, and so on) have the given characteristics fx=sin2x+3gx=3cos12x4hx=3sin12x5kx=3sec2x+mx=2csc2x23nx=3tanx2px=2cot12x+tx=3+2cosx Has a vertical shift upward from the parent graphFor Exercises 1-16, identify which functions shown here (f,g,h, and so on) have the given characteristics fx=sin2x+3gx=3cos12x4hx=3sin12x5kx=3sec2x+mx=2csc2x23nx=3tanx2px=2cot12x+tx=3+2cosx Has a vertical shift downward from the parent graph9PREFor Exercises 1-16, identify which functions shown here (f,g,h, and so on) have the given characteristics fx=sin2x+3gx=3cos12x4hx=3sin12x5kx=3sec2x+mx=2csc2x23nx=3tanx2px=2cot12x+tx=3+2cosx Has no x-interceptsFor Exercises 1-16, identify which functions shown here (f,g,h, and so on) have the given characteristics fx=sin2x+3gx=3cos12x4hx=3sin12x5kx=3sec2x+mx=2csc2x23nx=3tanx2px=2cot12x+tx=3+2cosx Has no y-interceptFor Exercises 1-16, identify which functions shown here (f,g,h, and so on) have the given characteristics fx=sin2x+3gx=3cos12x4hx=3sin12x5kx=3sec2x+mx=2csc2x23nx=3tanx2px=2cot12x+tx=3+2cosx Has a range of all real numbers13PREFor Exercises 1-16, identify which functions shown here (f,g,h, and so on) have the given characteristics fx=sin2x+3gx=3cos12x4hx=3sin12x5kx=3sec2x+mx=2csc2x23nx=3tanx2px=2cot12x+tx=3+2cosx Has a phase shift of 2For Exercises 1-16, identify which functions shown here (f,g,h, and so on) have the given characteristics fx=sin2x+3gx=3cos12x4hx=3sin12x5kx=3sec2x+mx=2csc2x23nx=3tanx2px=2cot12x+tx=3+2cosx Has a phase shift of 2For Exercises 1-16, identify which functions shown here (f,g,h, and so on) have the given characteristics fx=sin2x+3gx=3cos12x4hx=3sin12x5kx=3sec2x+mx=2csc2x23nx=3tanx2px=2cot12x+tx=3+2cosx Has no phase shiftFind the exact values or state that the expression is undefined. a. sin122 b. arcsin1 c. sin13Find the exact values. a. cos11 b. tan133 c. arctan0Use a calculator to approximate the function values in both radians and degrees. a. tan17.92 b. arccos27 c. sin10.81Use a calculator to approximate the degree measure (to 1 decimal place) or radian measure (to 4 decimal places) of the angle 6 subject to the given conditions. a. sin=37 and 180270 b. tan=83 and 2Find the exact values. a. coscos11 b. cos1cos43Find the exact value of sintan1125 .Find the exact value of cossin1211 .Write the expressiontansin1xx2+9 as an algebraic expression for x0 .For the construction of a house, a 16-ft by 6-ft wooden frame is made. Find the angle that the diagonal beam makes with the base of the frame. Round to the nearest tenth of a degree.Approximate each expression in radians, rounded to 4 decimal places. a. sec14 b. cos1512A function must be on its entire domain to have an inverse function.If 23,12 is on the graph of y=cosx , what is the related point on y=cos1x ?The graph of y=tan1x has two (horizontal/vertical) asymptotes represented by the equations and .The domain of y=arctanx is . The output is a real number (or angle in radians) between and .In interval notation, the domain of y=cos1x is . The output is a real number (or angle in radians) between and , inclusive.In interval notation, the domain ofy=sin1x is . The output is a real number (or angle in radians) between and , inclusive.For Exercises 7-12, find the exact value or state that the expression is undefined. (See Example 1) arcsin22For Exercises 7-12, find the exact value or state that the expression is undefined. (See Example 1) sin112For Exercises 7-12, find the exact value or state that the expression is undefined. (See Example 1) sin1For Exercises 7-12, find the exact value or state that the expression is undefined. (See Example 1) sin132For Exercises 7-12, find the exact value or state that the expression is undefined. (See Example 1) arcsin22For Exercises 7-12, find the exact value or state that the expression is undefined. (See Example 1) arcsin32For Exercises 13-16, find the exact value. sin132+sin112For Exercises 13-16, find the exact value. sin122sin11For Exercises 13-16, find the exact value. 2sin1223For Exercises 13-16, find the exact value. 2+3sin132For Exercises 17-28, find the exact value or state that the expression is undefined. (See Example 2) arccos22For Exercises 17-28, find the exact value or state that the expression is undefined. (See Example 2) tan13For Exercises 17-28, find the exact value or state that the expression is undefined. (See Example 2) cos1020PEFor Exercises 17-28, find the exact value or state that the expression is undefined. (See Example 2) cos12For Exercises 17-28, find the exact value or state that the expression is undefined. (See Example 2) arccos43For Exercises 17-28, find the exact value or state that the expression is undefined. (See Example 2) arctan33For Exercises 17-28, find the exact value or state that the expression is undefined. (See Example 2) cos12225PEFor Exercises 17-28, find the exact value or state that the expression is undefined. (See Example 2) arctan3327PE28PEFor Exercises 29-32, find the exact value. 3tan11+tan13For Exercises 29-32, find the exact value. cos122+cos112For Exercises 29-32, find the exact value. 2cos132tan133For Exercises 29-32, find the exact value. 3tan11+cos122For Exercises 33-36, use a calculator to approximate the function values in both radians and degrees. (See Example 3) a. cos138 b. tan125 c. arcsin0.05For Exercises 33-36, use a calculator to approximate the function values in both radians and degrees. (See Example 3) a. sin10.93 b. arccos0.17 c. arctan7435PE36PEFor Exercises 37-46, use a calculator to approximate the degree measure (to 1 decimal place) or radian measure (to 4 decimal places) of the angle subject to the given conditions. (See Example 4) cos=56 and 180270For Exercises 37-46, use a calculator to approximate the degree measure (to 1 decimal place) or radian measure (to 4 decimal places) of the angle subject to the given conditions. (See Example 4) sin=45 and 180270For Exercises 37-46, use a calculator to approximate the degree measure (to 1 decimal place) or radian measure (to 4 decimal places) of the angle subject to the given conditions. (See Example 4) tan=125 and 90180For Exercises 37-46, use a calculator to approximate the degree measure (to 1 decimal place) or radian measure (to 4 decimal places) of the angle subject to the given conditions. (See Example 4) cos=213 and 180270For Exercises 37-46, use a calculator to approximate the degree measure (to 1 decimal place) or radian measure (to 4 decimal places) of the angle subject to the given conditions. (See Example 4) sin=1219 and 90180For Exercises 37-46, use a calculator to approximate the degree measure (to 1 decimal place) or radian measure (to 4 decimal places) of the angle subject to the given conditions. (See Example 4) tan=715 and 18027043PEFor Exercises 37-46, use a calculator to approximate the degree measure (to 1 decimal place) or radian measure (to 4 decimal places) of the angle subject to the given conditions. (See Example 4) tan=95 and 245PEFor Exercises 37-46, use a calculator to approximate the degree measure (to 1 decimal place) or radian measure (to 4 decimal places) of the angle subject to the given conditions. (See Example 4) cos=117 and 322For Exercises 47-58, find the exact values. (See Example 5) sinsin122For Exercises 47-58, find the exact values. (See Example 5) arcsinsin53For Exercises 47-58, find the exact values. (See Example 5) sin1sin5450PEFor Exercises 47-58, find the exact values. (See Example 5) coscos123For Exercises 47-58, find the exact values. (See Example 5) arccoscos116For Exercises 47-58, find the exact values. (See Example 5) cos1cos43For Exercises 47-58, find the exact values. (See Example 5) coscos112For Exercises 47-58, find the exact values. (See Example 5) tan1tan23For Exercises 47-58, find the exact values. (See Example 5) tantan1257PEFor Exercises 47-58, find the exact values. (See Example 5) tan1tan6For Exercises 59-70, find the exact values. (See Examples 6-7) costan133For Exercises 59-70, find the exact values. (See Examples 6-7) sincos11261PEFor Exercises 59-70, find the exact values. (See Examples 6-7) sincos123For Exercises 59-70, find the exact values. (See Examples 6-7) sincos134For Exercises 59-70, find the exact values. (See Examples 6-7) sintan143For Exercises 59-70, find the exact values. (See Examples 6-7) sintan1166PEFor Exercises 59-70, find the exact values. (See Examples 6-7) cossin127For Exercises 59-70, find the exact values. (See Examples 6-7) costan1512For Exercises 59-70, find the exact values. (See Examples 6-7) tancos15670PEFor Exercises 71-76, write the given expression as an algebraic expression. It is not necessary to rationalize the denominator. (See Example 8) cossin1x25+x2 for x0 .For Exercises 71-76, write the given expression as an algebraic expression. It is not necessary to rationalize the denominator. (See Example 8) cotcos1x21x for x1 .For Exercises 71-76, write the given expression as an algebraic expression. It is not necessary to rationalize the denominator. (See Example 8) sintan1x for x0 .For Exercises 71-76, write the given expression as an algebraic expression. It is not necessary to rationalize the denominator. (See Example 8) tansin1x for x1 .75PEFor Exercises 71-76, write the given expression as an algebraic expression. It is not necessary to rationalize the denominator. (See Example 8) sincos1x225x for x5 .To meet the requirements of the Americans with Disabilities Act (ADA) a wheelchair ramp must have a slope of 1:12 or less. That is, for every 1in. of “rise,� there must be at least 12in. of “run.� (See Example 9) a. If a wheelchair ramp is constructed with the maximum slope, what angle does the ramp make with the ground? Round to the nearest tenth of a degree. b. If the ramp is 22ft long, how much elevation does the ramp provide? Round to the nearest tenth of a foot.A student measures the length of the shadow of the Washington Monument to be 620ft . If the Washington Monument is 555ft tall, approximate the angle of elevation of the Sun to the nearest tenth of a degree.A balloon advertising an open house is stabilized by two cables of lengths 20ft and 40ft tethered to the ground. If the perpendicular distance from the balloon to the ground is 103ft , what is the degree measure of the angle each cable makes with the ground? Round to the nearest tenth of a degree if necessary.A group of campers hikes down a steep path. One member of the group has an altimeter on his watch to measure altitude. If the path is 1250yd and the amount of altitude lost is 480yd , what is the angle of incline? Round to the nearest tenth of a degree.Navajo Tube Hill, a snow tubing hill in Utah, is 550ft long and has a 75-ft vertical drop. Find the angle of incline of the hill. Round to the nearest tenth of a degree.A ski run on Giant Steps Mountain in Utah is 1475m long. The difference in altitude from the beginning to the end of the run is 350m . Find the angle of the ski run. Round to the nearest tenth of a degree.83PE84PEShow that sec1x=cos11x for x1 .Show that csc1x=sin11x for x1.Show that secx1+csc1x=2 for x1 .88PE89PEFor Exercises 89-94, find the exact values. sec1291PE92PEFor Exercises 89-94, find the exact values. cot13For Exercises 89-94, find the exact values. cot11For Exercises 95-100, use a calculator to approximate each expression in radians, rounded to 4 decimal places. (See Example 10) sec174For Exercises 95-100, use a calculator to approximate each expression in radians, rounded to 4 decimal places. (See Example 10) csc165For Exercises 95-100, use a calculator to approximate each expression in radians, rounded to 4 decimal places. (See Example 10) csc1898PEFor Exercises 95-100, use a calculator to approximate each expression in radians, rounded to 4 decimal places. (See Example 10) cot1815For Exercises 95-100, use a calculator to approximate each expression in radians, rounded to 4 decimal places. (See Example 10) cot1247For Exercises 101-104, find the exact value if possible. Otherwise find an approximation to 4 decimal places or state that the expression is undefined. a. sin4 b. sin14 c. sin122102PEFor Exercises 101-104, find the exact value if possible. Otherwise find an approximation to 4 decimal places or state that the expression is undefined. a. cos16 b. cos6 c. cos132For Exercises 101-104, find the exact value if possible. Otherwise find an approximation to 4 decimal places or state that the expression is undefined. a. sin176 b. sin76 c. sin112105PEFor Exercises 105-108, find the inverse function and its domain and range. gx=6cosx4 for 0xFor Exercises 105-108, find the inverse function and its domain and range. hx=4+tanx For 2x2For Exercises 105-108, find the inverse function and its domain and range. kx=+sinx for 2x2A video camera located at ground level follows the liftoff of an Atlas V Rocket from the Kennedy Space Center. Suppose that the camera is 1000m from the launch pad. a. Write the angle of elevation from the camera to the rocket as a function of the rocket's height h . b. Without the use of a calculator, will the angle of elevation be less than 45 or greater than 45 when the rocket is 400m high? c. Use a calculator to find to the nearest tenth of a degree when the rocket's height is 400m,1500m , and 3000m .The effective focal length f of a camera is the distance required for the lens to converge light to a single focal point. The angle of view of a camera describes the angular range (either horizontally, vertically, or diagonally) that is imaged by a camera. a. Show that =2arctand2f where d is the dimension of the image sensor or film. b. A typical 35-mm camera has image dimensions of 24mm (vertically) by 36mm (horizontally). If the focal length is 50mm , find the vertical and horizontal viewing angles. Round to the nearest tenth of a degree.For Exercises 111-114, use the relationship given in the right triangle and the inverse sine, cosine, and tangent functions to write as a function of x in three different ways. It is not necessary to rationalize the denominator.For Exercises 111-114, use the relationship given in the right triangle and the inverse sine, cosine, and tangent functions to write as a function of x in three different ways. It is not necessary to rationalize the denominator.113PEFor Exercises 111-114, use the relationship given in the right triangle and the inverse sine, cosine, and tangent functions to write as a function of x in three different ways. It is not necessary to rationalize the denominator.For Exercises 115-120, find the exact solution to each equation. 2sin1x=0For Exercises 115-120, find the exact solution to each equation. 3cos1x=0For Exercises 115-120, find the exact solution to each equation. 6cos1x3=0118PEFor Exercises 115-120, find the exact solution to each equation. 4tan12x=120PE121PEExplain the flaw in the logic:cos4=22 . Therefore, cos122=4 .123PE124PEIn calculus, we can show that the area below the graph of fx=11+x2 , above the x- axis, and between the lines x=a and x=b for ab , is given by tan1btan1a a. Find the area under the curve between x=0 and x=1 . b. Evaluate f0 and f1 . c. Find the area of the trapezoid defined by the points 0,0,1,0,0,f0, and 1,f1 to confirm that your answer from part (a) is reasonable.126PEThe vertical viewing angle to a movie screen is the angle formed from the bottom of the screen to a viewer's eye to the top of the screen. Suppose that the viewer is sitting x horizontal feet from an IMAX screen 53 ft high and that the bottom of the screen is 10 vertical feet above the viewer's eye level. Let a be the angle of elevation to the bottom of the screen. a. Write an expression for tan . b. Write an expression for tan+ . c. Using the relationships found in parts (a) and (b), show that =tan163xtan110x .128PEa. Graph the functions y=tan1x and y=xx33+x55 on the window 2,2,4 by 2,2,1 . b. Graph the functionsy=tan1x and y=x1+x23x2 on the window 2,2,4 by 2,2,1 . c How do the functions in parts (a) and (b) compare for values of x taken close to 0 ?For Exercises 1-2, factor the expression completely. sin4xsin2xcos2xFor Exercises 1-2, factor the expression completely. 12tan2x+11tanx15For Exercises 3-4, find the LCD of the expressions. sinxtanx1;cosxtanx+14RE5REFor Exercises 5-6, simplify the expression. sec2x1sin2x7RE8REFor Exercises 7-14, verify the identity. 1cscx1+1csc+1=2tanxsecx10REFor Exercises 7-14, verify the identity. lncscx+lntanx=lnsecxFor Exercises 7-14, verify the identity. lntanxlnsinx+lncosx=013RE14REWrite 16x2 as a function of by making the substitution x=4cosfor02.16RE17REIn Exercises 17-26, use an addition or subtraction formula to find the exact value. cos10519RE20RE21REIn Exercises 17-26, use an addition or subtraction formula to find the exact value. tan1712In Exercises 17-26, use an addition or subtraction formula to find the exact value. cos40cos80sin40sin80In Exercises 17-26, use an addition or subtraction formula to find the exact value. tan2918+tan5361tan2918tan536In Exercises 17-26, use an addition or subtraction formula to find the exact value. tansin12853+cos1513In Exercises 17-26, use an addition or subtraction formula to find the exact value. sinarcco34+arcsin1227REFind the exact value for cos given sin=2129 for in Quadrant I and cos=2425 for in Quadrant III.For Exercises 29-32, verify the identity. cosxycosxsiny=coty+tanxFor Exercises 29-32, verify the identity. sinxycosxcosy=tanxtany31REFor Exercises 29-32, verify the identity. sinx+4cosx+4=2sinxWrite 3sinxcosx in the form ksinx+ for 02 .Write 3cosx4sinx in the form ksinx+a for 02 . Round to 3 decimal places.For Exercises 35-38, verify the identity. 2tanx1+tan2x=sin2x36REFor Exercises 35-38, verify the identity. sin2x2cos2x2=cosx38REWrite 16cos4x in terms of first powers of cosine.40REFor Exercises 41-44, use the given information to find the exact value of each expression. a.sin2b.cos2c.sin2d.cos2 sin=513,2For Exercises 41-44, use the given information to find the exact value of each expression. a.sin2b.cos2c.sin2d.cos2 cos=45,3243RE44RE45REFor Exercises 45-46, use the given information to find the exact value of each expression. a.tan2b.tan2 sin=38,cos047RE48REFor Exercises 47-50, write the product as a sum or difference. sin3xcos6xFor Exercises 47-50, write the product as a sum or difference. cos10xcos5x51REFor Exercises 51-54, write each expression as a product. sin10x+sin2x53RE54REFor Exercises 55-56, use a product-to-sum formula to find the exact value. sin37.5cos7.5For Exercises 55-56, use a product-to-sum formula to find the exact value. cos54cos51257RE58REFor Exercises 59-60, use the sum-to-product formulas to verify the identity. cos4+tcos4t=2sint60REFor Exercises 61-62, verify the identity. sin4xsin2xcos4xcos2x=cot3xFor Exercises 61-62, verify the identity. sin3x+sin5x+sin8x=4sin4xcos5x2cos3263REFor Exercises 63-70, a. Write the solution set for the general solution. b. Write the solution set on the interval 0,2 . 42=2cosx+32For Exercises 63-70, a. Write the solution set for the general solution. b. Write the solution set on the interval 0,2 . 3sec2x=466RE67REFor Exercises 63-70, a. Write the solution set for the general solution. b. Write the solution set on the interval 0,2 . cos3x=12For Exercises 63-70, a. Write the solution set for the general solution. b. Write the solution set on the interval 0,2 . tanx2=1For Exercises 63-70, a. Write the solution set for the general solution. b. Write the solution set on the interval 0,2 . sinx2=3271REFor Exercises 71-88, solve the equations on the interval 0,2 . tanx2=3373REFor Exercises 71-88, solve the equations on the interval 0,2 . cos=3475REFor Exercises 71-88, solve the equations on the interval 0,2 . 4cosx=777REFor Exercises 71-88, solve the equations on the interval 0,2 . 6cscx2+11cscx2=0For Exercises 71-88, solve the equations on the interval 0,2 . 17cos2x+4cosx1=0For Exercises 71-88, solve the equations on the interval 0,2 . 10sin2x3sinx4=0For Exercises 71-88, solve the equations on the interval 0,2 . 2cos2x5sinx+1=082RE83REFor Exercises 71-88, solve the equations on the interval 0,2 . sinx=cos2x85RE86REFor Exercises 71-88, solve the equations on the interval 0,2 . cosx+1=sinx88RE89RE90RE91RE92RE93REFor Exercises 1-2, simplify the expression. 1cot2+1+1tan2+12T3TFor Exercises 3-8, verify the identity. cotxtanxcotx+tanx=cos2x5T6T7T8T9TWrite 8cosx15sinx in the form ksinx+ for 02 . Round to 3 decimal places.11TFor Exercises 12-17, find the exact value. sin250cos10cos250sin1013TFor Exercises 12-17, find the exact value. cosarctan3+arcsin4515T16TFor Exercises 12-17, find the exact value. sin555+sin10518T19TGiven tan=158 and 32 find the exact function values. a. sin2 b. cos2 c. tan2For Exercises 21-22 a. Write the solution set for the general solution. b. Write the solution set on the interval 0,2 . 3tanx+5=622TFor Exercises 23-30, solve the equation on the interval [0,2) . 2cos3xcosx=0