WISKUNDE
Graad 11
NOG OEFENINGE
Reduksie formules.
  
        Vraag  1
     Skryf die volgende as verhoudings van net θ :

       1.1  sin (180° − θ)                 [ A 1.1 ]
       1.2  cos (180° − θ)                 [ A 1.2 ]
       1.3  tan (360° − θ)                 [ A 1.3 ]
       1.4  sec (360° + θ)                [ A 1.4 ]
       1.5  cosec (360° − θ)             [ A 1.5 ]
       1.6  cot (180° + θ)                [ A 1.6 ]
       1.7  tan (180° − θ)                [ A 1.7 ]
       1.8  sin (180° + θ)                [ A 1.8 ]
       1.9  cos (360° − θ)                 [ A 1.9 ]
       1.10  cosec (180° − θ)           [ A 1.10 ]
       1.11  cot (360° − θ)               [ A 1.11 ]
       1.12  tan (180° + θ)              [ A 1.12 ]
       1.13  sin (90° − θ)                 [ A 1.13 ]
       1.14  cos (90° + θ)                [ A 1.14 ]
       1.15  tan (270° − θ)               [ A 1.15 ]
       1.16  cot (270° + θ)              [ A 1.16 ]
       1.17  cosec (90° − θ)             [ A 1.17 ]
       1.18  sec (90° + θ)                [ A 1.18 ]
       1.19  tan (90° + θ)                [ A 1.19 ]
       1.20  sec (270° + θ)              [ A 1.20 ]
       1.21  sin (270° − θ)               [ A 1.21 ]
       1.22  cos (270° + θ)              [ A 1.22 ]
       1.23  sin (90° + θ)                 [ A 1.23 ]
       1.24  cos (90° − θ)                 [ A 1.24 ]

        Vraag  2
     Skryf elk van die volgende as 'n verhouding van 20° :
     functions :

       2.1  sin 70°                      [ A 2.1 ]
       2.2  cos 110°                   [ A 2.2 ]
       2.3  cosec 200°               [ A 2.3 ]
       2.4  sec 340°                   [ A 2.4 ]
       2.5  cot 250°                   [ A 2.5 ]
       2.6  tan 380°                   [ A 2.6 ]
       2.7  sin 200°                   [ A 2.7 ]
       2.8  cos 160°                   [ A 2.8 ]
       2.9  tan 290°                   [ A 2.9 ]
       2.10  cosec 70°               [ A 2.10 ]
       2.11  cot 110°                 [ A 2.11 ]
       2.12  sec 250°                 [ A 2.12 ]
       2.13  cot 200°                 [ A 2.13 ]
       2.14  cosec 250°             [ A 2.14 ]
       2.15  sin 110°                 [ A 2.15 ]
       2.16  cos 70°                   [ A 2.16 ]
       2.17  tan 110°                 [ A 2.17 ]
       2.18  sec 70°                   [ A 2.18 ]

        Vraag  3
     Skryf elk van die volgende as 'n verhouding van 40° :

       3.1  cos 50°                           [ A 3.1 ]
       3.2  cosec 320°                     [ A 3.2 ]
       3.3  cot 310°                         [ A 3.3 ]
       3.4  tan 130°                        [ A 3.4 ]
       3.5  sin 230°                         [ A 3.5 ]
       3.6  sec 400°                         [ A 3.6 ]
       3.7  cos 310°                         [ A 3.7 ]
       3.8  sec 230°                         [ A 3.8 ]
       3.9  cosec 130°                     [ A 3.9 ]
       3.10  sin 220°                       [ A 3.10 ]
       3.11  cos 130°                       [ A 3.11 ]
       3.12  tan 230°                       [ A 3.12 ]
       3.13  cot 50°                         [ A 3.13 ]
       3.14  cosec 310°                   [ A 3.14 ]
       3.15  sec 140°                       [ A 3.15 ]
       3.16  cos 230°                       [ A 3.16 ]
       3.17  sin 320°                       [ A 3.17 ]
       3.18  tan 140°                       [ A 3.18 ]

        Vraag  4
                 Skryf elk van die volgende as 'n verhouding
                 van 'n skerp hoek :

       4.1  sin 155°                                [ A 4.1 ]

       4.2  cosec 222°                            [ A 4.2 ]

       4.3  tan 145°                                [ A 4.3 ]

       4.4  cot 213°                                [ A 4.4 ]

       4.5  sec 298°                                [ A 4.5 ]

       4.6  cosec 124°                            [ A 4.6 ]

       4.7  sin 138°                                [ A 4.7 ]

       4.8  sec 318°                                [ A 4.8 ]

       4.9  cot 118°                                [ A 4.9 ]

       4.10  cosec 346°                          [ A 4.10 ]

       4.11  cot 288°                              [ A 4.11 ]

       4.12  tan 113°                              [ A 4.12 ]

       4.13  cos 308°                              [ A 4.13 ]

       4.14  sin 150°                              [ A 4.14 ]

       4.15  sec 218°                               [ A 4.15 ]

       4.16  cot 128°                               [ A 4.16 ]

       4.17  cosec 232°                          [ A 4.17 ]

       4.18  cos 142°                              [ A 4.18 ]

        Vraag  5
                 As sin 23° = p, skryf die volgnde
                 in terme van p :

       5.1  cos 67°                                   [ A 5.1 ]

       5.2  sin 157°                                 [ A 5.2 ]

       5.3  cos 293°                                [ A 5.3 ]

       5.4  sin 203°                                [ A 5.4 ]

       5.5  cos 113°                                [ A 5.5 ]

       5.6  sin 337°                                 [ A 5.6 ]

       5.7  cos 23°                                   [ A 5.7 ]

       5.8  cos 247°                                 [ A 5.8 ]

       5.9  cosec 23°                               [ A 5.9 ]

       5.10  tan 23°                                 [ A 5.10 ]

        Vraag  6
            Vereenvoudig die volgende so ver as moontlik :

      6.1  sec (180° + x)
            . cosec (180° ─ x) . cos (360° ─ x) . sin (180° + x)
                                                                                 [ A 6.1 ]

       6.2  tan (180° + x) . cos (360° ─ x) . sec (90° + x)
                                                                                 [ A 6.2 ]

       6.3  tan (180° + x) − tan (180° − x) +
                                  tan (360° + x) − tan (360° − x)
                                                                                 [ A 6.3 ]

       6.4  sin (180° ─ x) .cos (90° ─ x) ─
                                   cos (180° + x) . cos (360° ─ x)
                                                                                 [ A 6.4 ]

       6.5  sin (90° + x) . cos (90° ─ x) +
                                  cos (90° + x) . sin (90° ─ x)
                                                                                 [ A 6.5 ]

    
$$ \hspace*{2 mm}\mathrm{6.6\kern3mm\frac{sin\ (360° −x) \bold{.} tan\ (180° + x)}{cos\ x\ \bold{.}\ cot\ (90° − x)\bold{.}\ sec\ (180° + x)}\kern2mm\ } $$

                                                                                 [ A 6.6 ]

    
$$ \hspace*{2 mm}\mathrm{6.7\kern3mm\frac{cos\ (180° + θ)\ \bold{.}\ sin\ (180° − θ)}{cos\ (360° − θ)\ \bold{.}\ tan\ (360° − θ)}\kern2mm\ } $$          [ A 6.7 ]

    
$$ \hspace*{2 mm}\mathrm{6.8\kern3mm\frac{sin\ (360° + x)\ ÷ cos\ (90° − x)\ ÷ sin\ (180° − x)}{tan\ (180° + x)\ ÷ tan\ (360° + x)\ ÷ cos\ (270° − x)}\kern2mm\ } $$

                                                                                 [ A 6.8 ]

    
$$ \hspace*{2 mm}\mathrm{6.9\kern3mm\frac{cos^2\ (180° − x)}{sin\ (360° − x)\ \bold{.}\ sin\ (90° − x)}\kern2mm\ } $$              [ A 6.9 ]

    
$$ \hspace*{2 mm}\mathrm{6.10\kern3mm\frac{cos\ (90° + α)\ \bold{.}\ sin\ (−\ α)}{sin\ (180° + α)\ \bold{.}\ tan\ (360° − α)}\kern2mm\ } $$        [ A 6.10 ]

    
$$ \hspace*{2 mm}\mathrm{6.11\kern3mm\frac{cos\ (−\ β)\ \bold{.}\ sin\ (180° + β)}{tan\ (β − 180°)\ \bold{.}\ sin\ (β + 90°)}\kern2mm\ } $$           [ A 6.11 ]

    
$$ \hspace*{2 mm}\mathrm{6.12\kern3mm\frac{sin\ (−\ α)\ \bold{.}\ sec\ (−\ α)\ \bold{.}\ tan\ (18° − α)}{sec\ (360° −\ α)\ \bold{.}\ cos\ (270° −\ α)\ \bold{.}\ tan\ (180° + α) }\kern2mm\ } $$

                                                                                 [ A 6.12 ]

        Vraag  7
             Vereenvoudig die volgende sonder die
             gebruik van 'n sakrekenaar :

    
$$ \hspace*{2 mm}\mathrm{7.1\kern3mm\ sin\ 320°\ .\ tan\ 140°\ .\ cot\ 220°\ .\ sec\ 130°\kern2mm\ } $$
                                                                                 [ A 7.1 ]
    
$$ \hspace*{2 mm}\mathrm{7.2\kern3mm\ sin\ 200°\ .\ sec\ 110°\ + cos\ 160°\ .\ cosec\ 110°\kern2mm\ } $$
                                                                                 [ A 7.2 ]
    
$$ \hspace*{2 mm}\mathrm{7.3\kern3mm\ cos\ 290°\ .\ tan\ 200°\ −\ cos\ 160°\kern2mm\ } $$
                                                                                 [ A 7.3 ]
    
$$ \hspace*{2 mm}\mathrm{7.4\kern3mm\ cos\ 63°\ .\ cosec\ 207°\ .\ cot\ 27°\kern2mm\ } $$         [ A 7.4 ]
    
$$ \hspace*{2 mm}\mathrm{7.5\kern3mm\ sin\ 125°\ .\ cot\ 145°\ .\ cosec\ 55°\ .\ sin\ 215°\kern2mm\ } $$
                                                                                 [ A 7.5 ]
    
$$ \hspace*{2 mm}\mathrm{7.6\kern3mm\frac{cosec\ 110°}{2\ sec\ 20°}\kern2mm\ } $$                                             [ A 7.6 ]
    
$$ \hspace*{2 mm}\mathrm{7.7\kern3mm\frac{tan\ 340°\ .\ sin\ 72°}{cos\ 18°\ .\ cot\ 70°}\kern2mm\ } $$                                 [ A 7.7 ]
    
$$ \hspace*{2 mm}\mathrm{7.8\kern3mm\frac{sec\ 185°\ .\ sin\ 78°}{cos\ 348°\ .\ sec\ 175°}\kern2mm\ } $$                             [ A 7.8 ]
    
$$ \hspace*{2 mm}\mathrm{7.9\kern3mm\frac{3\ tan\ 65°\ .\ cosec\ 160°}{4\ cosec\ 200°\ .\ cot\ 25°}\kern2mm\ } $$                         [ A 7.9 ]
    
$$ \hspace*{2 mm}\mathrm{7.10\kern3mm\frac{3\ cot\ 27°\ .\ tan\ 153°}{sin\ 27°\ .\ cos\ 117°\ −\ cosec\ 207°\ .\ sin\ 333°}\kern2mm\ } $$

                                                                                [ A 7.10 ]
    
$$ \hspace*{2 mm}\mathrm{7.11\kern3mm\frac{1}{cot\ 52°}\ \times \ cosec\ 128°\ + sec\ 232°\ .\ cos\ 142°\kern2mm\ } $$
                                                                                [ A 7.11 ]
    
$$ \hspace*{2 mm}\mathrm{7.12\kern3mm\frac{sin\ 108°\ .\ tan\ 72°\ .\ sin\ 162°}{tan\ 252°\ .\ cos\ (−\ 72°)\ .\ sin\ (−\ 72°)}\kern2mm\ } $$
                                                                                [ A 7.12 ]

  
  
  
  
  
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