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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