Solution to Chapter 6 Problems

1) a) 210   sin210 = sin(π+30) = -sin30 = -1/2.

                        cos210 = cos(π+30) = -cos30 = -3/2.

                        tan210 = tan(π+30) = + tan30 = +3/3.

                   cot210 = cot(π+30) = + cot30 = +3.

   b) 315     sin315 = sin(-45) = -sin45 = -2/2.

                        cos315 = cos(-45) = +cos45 = +2/2.

                        tan315 = tan(-45) = - tan45 = -1.

                   cot315 = cot(-45) = - cot45 = -1.

   c) 120     sin120 = sin(π-60) = +sin60 = +3/2.

                        cos120 = cos(π-60) = -cos60 = -1/2.

                        tan120 = tan(π-60) = - tan60 = -3.

                   cot120 = cot(π-60) = - cot60 = -3/3.

   d) 4π/3     sin(4π/3) = sin (π + π/3) = -sin(π/3) = -3/2.

                       cos(4π/3) = cos (π + π/3) = -cos(π/3) = -1/2.

                       tan(4π/3) = tan (π + π/3) = + tan(π/3) = +3.

                  cot(4π/3) = cot (π + π/3) = + cot(π/3) = +3/3.

  e) 11π/6     sin(11π/6) = sin ( - π/6) = -sin (π/6) = -1/2.

                       cos(11π/6) = cos ( - π/6) = +cos (π/6) = +3/2.

                       tan(11π/6) = tan ( - π/6) = - tan (π/6) = -3/3.

                  cot(11π/6) = cot ( - π/6) = - cot (π/6) = -3.

  f) 3π/4     sin(3π/4) = sin (π - π/4) = +sin (π/4) = +2/2.

                       cos(3π/4) = cos (π - π/4) = -cos (π/4) = -2/2.

                       tan(3π/4) = tan (π - π/4) = - tan (π/4) = -1.

                  cot(3π/4) = cot (π - π/4) = - cot (π/4) = -1.

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2) Numerator:    sin150 cos30 - 2sin30 tan120 =

                    sin(π - 30) cos30 - 2sin30 tan(π - 60) =

                    sin30 cos30  -  2sin30[-tan60] =  

                    (1/2)(3/2)  -   2(1/2)( - 3) = 5√3/4

  Denominator2cot60 - tan120  + cot330 =  

                              2cot60   - tan(π - 60) + cot(2π - 30) =  

                              2cot60   -[- tan60 ]  - cot30 =  

                              2cot60 + tan60  - cot30 =  

                            2(√3/3) +  3    3  =  2√3/3  

Now,    (Num.)/(Denom.) =  ( 5√3/4) /2√3/3) = 15/8                        

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3) Numerator:    2sin18 - sin(-18) + sin162 - 2sin198 =  

                              2sin18 - [-sin18]  + sin(180 - 18) -2sin(180 + 18) =  

                              2sin18 + sin18  +  sin18  -2[-sin18]  =   6 sin18

   Denominator:  cos(-72) + 3cos108 + sin342 =  

                               cos72   + 3 cos ( 90 + 18) + sin (360 - 18) =  

                                sin18  - 3 sin18 - sin18  =  -3sin18

Now,    (Num.)/(Denom.) =  ( 6 sin18 ) / ( -3sin18) = -2                     

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4)  sin(π -a)sin(2π +a) + cos(π/2 -a)cos(π/2 +a) + cos2 (π + a) =

                                                             1/(1 + tan2a).

Solution:  (sina)(sina)  + (sina)[-sina]  + [-cosa]2 =  

                                                                  cos2a  =  

                                                     1/(1 + tan2a).

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 5)              tan(π + x)cot(x - π) - cos(2π - x)cos(2π + x) = sin2x.

Solution:  (tanx)[-cot(π - x)]  - cosx cos =

                   (tanx) (cotx- cos2x =

                            1  - cos2x =

                                sin2x.

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6)                    sin(π - a)sin(π/2 - a) [ tan(π + a) - cot(2π - a) ] = 1.

Solution:            sina cosa [ tana + cota ] =

                            sina cosa [ sina/cosa  +  cosa/sina ] =

Multiplying sina cosa through, we get :

                                            sin2a + cos2a =

                                                        1.

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7)    sin(π/2 + x)cos(π/2 - x) + sin(x - π)cos(x - 2π) + tan(-x)tan(π/2 + x) = 1.

Solution:   cosx sinx   -  sin(π - x) cos( 2π - x)   +  [-tanx] [-cotx] =

                    sinx cosx   - sinx cosx     +    1  =

                                                                1.

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