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All Textbook Solutions for Precalculus

For Exercises 23-30, solve the equation on the interval [0,2) . 6csc2x17cscx+10=0 For Exercises 23-30, solve the equation on the interval [0,2) . 6sin2x5cosx2=0 26T 27T 28T 29T For Exercises 23-30, solve the equation on the interval [0,2) . sinx+sin3xsin2x=0 31T For a projectile launched from ground level at an angle of elevation with an initial velocity v0 , the maximum horizontal range is given by xmax=v02sin2g , where g is the acceleration due to gravity g=32ft/sec2org=9.8m/sec2 . If a toy rocket is launched from the ground with an initial velocity of 50ft/sec and lands 73 ft from the launch point, find the angle of elevation of the rocket at launch. Round to the nearest tenth of a degree.1CRE 2CRE 3CRE 4CRE 5CRE 6CRE 7CRE 8CRE 9CRE 10CRE 11CRE 12CRE 13CRE 14CRE 15CRE 16CRE Given fx=log3x, a. Write the domain and range in interval notation. b. Write an equation of the asymptote. c. Find the intercepts. d. Determine the intervals over which f is increasing, decreasing, and constant.18CRE 19CRE 20CRE Simplify. Write the final form with no fractions. tanxcos2xsecx Simplify. Write the final form with no fractions. cossin+sin1+cos Simplify. Write the final form with no fractions. 1sec2ttanttantsect Verify that the equation is an identity. sinxcotxcosx=1 5SP 6SP Verify that the equation is an identity. lncotx+lnsinx=lncosx 8SP 9SP The value tan is the quotient of and , and cot is the quotient of .2PE Given the Pythagorean identitysin2x+cos2x=1 , write two alternative forms that involve the difference of squares.Given the Pythagorean identity tan2x+1=sec2x , write two alternative forms that involve the difference of squares.5PE An equation that is true for all values of the variable except where the individual expressions are not defined is called an .For Exercises 7-10, find the least common denominator. Then rewrite each expression with the new denominator. a. 1a;1b b. 1cosx;1sinx 8PE For Exercises 7-10, find the least common denominator. Then rewrite each expression with the new denominator. a. 11+a;ab b. 11+sinx;sinxcosx 10PE For Exercises 11-12, multiply. a. a+b2 b. cosx+cotx2 12PE 13PE 14PE 15PE 16PE 17PE For Exercises 13-20, factor each expression. a. a22ab+b2 b. sin2x2sinxtanx+tan2x 19PE 20PE 21PE For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) cosxtanxcscx For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) sinxsecxtanx For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) sinxsecxcotx For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) tan2xcotxcscx For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) csc2xtan2xsecx For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) tan2xsinxsec2x For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) tanxsinxsec2x For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) cos2x+1sinx+sinx For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) sin2x+1cos2x+1 For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) cscsincsc For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) cossecsec For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) sin2x+2sinx+1sin2x+sinx For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) sinxsec2xsinxsinxsecx+sinx For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) cos2x2cosx+1sinxcosxsinx For Exercises 21-36, simplify the expression. Write the final form with no fractions. (See Example 1-3) tan2x+2tanx+1sinxtanx+sinx For Exercises 37-40, verify that the equation is an identity. (See Example 4) sinx+cscx=cotxcosx For Exercises 37-40, verify that the equation is an identity. (See Example 4) sinx+cotxcosx=cscx For Exercises 37-40, verify that the equation is an identity. (See Example 4) cotxcotx+tanx=csc2x For Exercises 37-40, verify that the equation is an identity. (See Example 4) csc2xtanxcotx=sec2x For Exercises 41-46, verify that the equation is an identity. (See Example 5) secxtanxtanxsecx=cosxcotx For Exercises 41-46, verify that the equation is an identity. (See Example 5) cotxcscxcscxcotx=sinxtanx For Exercises 41-46, verify that the equation is an identity. (See Example 5) 11+sint=11sint=2sec2t For Exercises 41-46, verify that the equation is an identity. (See Example 5) 1sect11sect+1=2cot2t For Exercises 41-46, verify that the equation is an identity. (See Example 5) 11+secx11secx=2cotxcscx For Exercises 47-52, verify that the equation is an identity. (See Example 5) 11cscx11+cscx=2tanxsecx For Exercises 47-52, verify that the equation is an identity. (See Example 6) sincsccot=1+cos For Exercises 47-52, verify that the equation is an identity. (See Example 6) costan+sec=1sin For Exercises 47-52, verify that the equation is an identity. (See Example 6) 1cosx+sinxcosx=1sinxcos3x For Exercises 47-52, verify that the equation is an identity. (See Example 6) 1sinxsin2x=1+sinxsinxcos2x For Exercises 47-52, verify that the equation is an identity. (See Example 6) secx+1secx1=(cscx+cotx)2 For Exercises 47-52, verify that the equation is an identity. (See Example 6) sinx+cosxsinxcosx=1+2sinxcosx12cos2x For Exercises 53-56, verify that the equation is an identity. (See Example 7) lncostlncott=lnsint For Exercises 53-56, verify that the equation is an identity. (See Example 7) lncottlntant=2lncott For Exercises 53-56, verify that the equation is an identity. (See Example 7) lnsec+tan=lnsectan For Exercises 53-56, verify that the equation is an identity. (See Example 6) lncsc+cot=lncsccot For Exercises 57-60, simplify each side of the equation independently to reach a common equivalent expression. (See Example 8) cosx1sinx=secx+tanx For Exercises 57-60, simplify each side of the equation independently to reach a common equivalent expression. (See Example 8) 1+tan2xtanx=cosxsinxsin3x For Exercises 57-60, simplify each side of the equation independently to reach a common equivalent expression. (See Example 8) cot2tcot2t=csc2tcos2tcos2t For Exercises 57-60, simplify each side of the equation independently to reach a common equivalent expression. (See Example 8) tan2t+sin2t=sec2tcos2t For Exercises 61-68, write the given algebraic expression as a function of , where 02, by making the given substitution. (See Example 9) x2+25; Substitutex=5tan .For Exercises 61-68, write the given algebraic expression as a function of , where 02, by making the given substitution. (See Example 9) 49x2; Substitute x=7sin .For Exercises 61-68, write the given algebraic expression as a function of , where 02, by making the given substitution. (See Example 9) 16x12; Substitute x=4sin+1 .For Exercises 61-68, write the given algebraic expression as a function of , where 02, by making the given substitution. (See Example 9) x+2236; Substitute x=6sec2 .For Exercises 61-68, write the given algebraic expression as a function of , where 02, by making the given substitution. (See Example 9) 9x2x ; Substitute x=3sec .For Exercises 61-68, write the given algebraic expression as a function of , where 02, by making the given substitution. (See Example 9) x2+16x ; Substitute x=4tan .For Exercises 61-68, write the given algebraic expression as a function of , where 02, by making the given substitution. (See Example 9) 149x213/2; Substitute x=17sec .68PE 69PE 70PE 71PE 72PE 73PE For Exercises 73-104, verify that the equation is an identity. 1sin21+tan2=1 For Exercises 73-104, verify that the equation is an identity. secxtanxsinx=sec2x For Exercises 73-104, verify that the equation is an identity. secxcotxsinx=csc2x For Exercises 73-104, verify that the equation is an identity. secxsinxcscxcosx=tan2x For Exercises 73-104, verify that the equation is an identity. sinxcotxcosxtanx=cotx 79PE For Exercises 73-104, verify that the equation is an identity. 2tan4x+7tan2x+5sec2x=2tan2x+5 81PE For Exercises 73-104, verify that the equation is an identity. cscxcosxcscxsinx=secx 83PE For Exercises 73-104, verify that the equation is an identity. csct+cott2=1+cost1cost 85PE 86PE 87PE For Exercises 73-104, verify that the equation is an identity. tanxsecx1=secx+1tanx 89PE 90PE 91PE 92PE 93PE 94PE 95PE 96PE For Exercises 73-104, verify that the equation is an identity. cosxtanxsecxcotx=cotxcosx 98PE 99PE 100PE 101PE For Exercises 73-104, verify that the equation is an identity. log100sinx=2sinx 103PE 104PE 105PE 106PE 107PE 108PE 109PE 110PE 111PE 112PE 113PE 114PE 115PE 116PE 117PE 118PE Find the exact values. a. sin195 b. cos512 Find the exact value of the expression. cos99cos36sin99sin36 3SP 4SP Find the exact value of tan165 .Verify the identity. cos32=sin Verify the identity. sinx+ysinxy=2cosxsiny Write 4sinx+3cosx in the form ksinx+ .1PE 2PE Fill in the boxes to complete each identity. tanu+v=1tanutanv;tanuv=tanutanv 4PE 5PE 6PE For Exercises 7-18, use an addition or subtraction formula to find the exact value. cos165 For Exercises 7-18, use an addition or subtraction formula to find the exact value. sin512 For Exercises 7-18, use an addition or subtraction formula to find the exact value. sin12 For Exercises 7-18, use an addition or subtraction formula to find the exact value. tan1912 For Exercises 7-18, use an addition or subtraction formula to find the exact value. cos1912 For Exercises 7-18, use an addition or subtraction formula to find the exact value. sin15 For Exercises 7-18, use an addition or subtraction formula to find the exact value. tan712 For Exercises 7-18, use an addition or subtraction formula to find the exact value. cos712 For Exercises 7-18, use an addition or subtraction formula to find the exact value. sin105 For Exercises 7-18, use an addition or subtraction formula to find the exact value. tan105 For Exercises 7-18, use an addition or subtraction formula to find the exact value. tan15 For Exercises 7-18, use an addition or subtraction formula to find the exact value. cos195 For Exercises 19-26, use an addition or subtraction formula to find the exact value. sin140cos20cos140sin20 For Exercises 19-26, use an addition or subtraction formula to find the exact value. cos3518cos518+sin3518sin518 For Exercises 19-26, use an addition or subtraction formula to find the exact value. tan54tan121tan54tan12 For Exercises 19-26, use an addition or subtraction formula to find the exact value. sin3536cos1318cos3536sin1318 For Exercises 19-26, use an addition or subtraction formula to find the exact value. cos23cos76sin23sin76 For Exercises 19-26, use an addition or subtraction formula to find the exact value. tan15tan451+tan15tan45 For Exercises 19-26, use an addition or subtraction formula to find the exact value. cos200cos25sin200sin25 For Exercises 19-26, use an addition or subtraction formula to find the exact value. sin109cos718+cos109sin718 For Exercises 27-32, find the exact value for the expression under the given conditions. cos+;sin=35 for in Quadrant III and cos=34 for in Quadrant II.For Exercises 27-32, find the exact value for the expression under the given conditions. sin+;sin=38 for in Quadrant II and cos=1213 for in Quadrant IV.For Exercises 27-32, find the exact value for the expression under the given conditions. tan+;sin=817 for in Quadrant II and cos=941 for in Quadrant III.30PE 31PE 32PE 33PE 34PE 35PE 36PE 37PE For Exercises 33-40, find the exact value. sincos11213+tan143 39PE 40PE 41PE 42PE 43PE For Exercises 41-62, verify the identity. cosx=cosx 45PE 46PE 47PE For Exercises 41-62, verify the identity. cos(xy)cos(x+y)=2sinxsiny 49PE 50PE 51PE For Exercises 41-62, verify the identity. cosx=4cosx4=2sinx 53PE 54PE 55PE For Exercises 41-62, verify the identity. sinsin+=tantantan+tan 57PE 58PE 59PE 60PE 61PE 62PE 63PE For Exercises 63-66, a. Write the given expression in the form ksinx+for02. Round to 3 decimal places. (See Example 8) b. Verify the result from part (a) by applying the sum formula for sine. 15sinx+8cosx 65PE 66PE 67PE 68PE 69PE 70PE 71PE a. Is it true that cos245=2cos45 ? b. Expand cos245 as cos45+45 using the sum formula for cosine. Then simplify the result.Derive cosu+v=cosucosvsinusinv by using the identity for cosuv and the odd and even function identities for sine and cosine.Derive tanuv=tanutanv1+tanutanv by using the identity for tanu+v and the odd function identity for tangent.Derive sinu+v=sinucosv+cosusinv 76PE For fx=sinx , show that fx+hfxh=sin1coshh+cosxsinhh Forfx=cosx , show that fx+hfxh=cosx1coshhsinxsinhh 79PE 80PE 81PE 82PE 83PE Describe the pattern for the expansions of cosu+v and cosuv .85PE 86PE 87PE 88PE 89PE 90PE 91PE 92PE 93PE 94PE 95PE 96PE 97PE Suppose that ABC contains no right angle. Show tanA+tanB+tanC=tanAtanBtanC .Let L be a line defined by y=mx+b with a positive slope, and let be the acute angle formed by L and the horizontal. Let px1,mx1+b and Qx2,mx2+b be arbitrary points on L . Show that m=tan .100PE 101PE 102PE Given that sin=45 for in Quadrant II, find the exact function values. a. sin2 b. cos2 c. tan2 2SP 3SP Use the half-angle formula to find the exact value of sin165 .Show that 1cos1+cos=sin1+cos.6SP 1PE 2PE 3PE 4PE From the relationship sin2=1cos22, it follows that sinu2 = .From the relationship cos2=1+cos22 , it follows that cos u2 = .For Exercises 7-14, use the given information to find the exact function values. a.sin2b.cos2c.tan2 sin=1213, in Quadrant IV For Exercises 7-14, use the given information to find the exact function values. a.sin2b.cos2c.tan2 sin=47 , tan0 For Exercises 7-14, use the given information to find the exact function values. a.sin2b.cos2c.tan2 cos=55,sin0 For Exercises 7-14, use the given information to find the exact function values. a.sin2b.cos2c.tan2 cos=58, in Quadrant III For Exercises 7-14, use the given information to find the exact function values. a.sin2b.cos2c.tan2 cos=52, in Quadrant III For Exercises 7-14, use the given information to find the exact function values. a.sin2b.cos2c.tan2 cos=26,cos0 For Exercises 7-14, use the given information to find the exact function values. a.sin2b.cos2c.tan2 sec=3712, in Quadrant IV For Exercises 7-14, use the given information to find the exact function values. a.sin2b.cos2c.tan2 csc=2524,cos0 For Exercises, 15-20, find the exact value of the expression. 2tan67.51tan267.5 For Exercises, 15-20, find the exact value of the expression. 2tan151tan215 For Exercises, 15-20, find the exact value of the expression. 2sin12cos12 For Exercises, 15-20, find the exact value of the expression. 2sin22.5cos22.5 For Exercises, 15-20, find the exact value of the expression. cos28sin28 For Exercises, 15-20, find the exact value of the expression. 2cos21651 21PE 22PE 23PE 24PE 25PE For Exercises, 21-34, verify the identity. sin3xsin2x=2cosx12secx 27PE 28PE

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