imath - more exercises, re. the reduction formulae of trigonometry die trigonometriese reduksie formules.

WISKUNDE
GRAAD 11
NOG OEFENINGE

Reduksie Formules.

MATHEMATICS
GRADE 11
MORE EXERCISES

Reduction Formulae.

Vraag / Question 1

                 Skryf die volgende as verhoudings van net θ :

             1.1    sin (180° − θ)                                 [ 1.1 ]

             1.3    tan (360° − θ)                                [ 1.3 ]

             1.5    cosec (360° − θ)                           [ 1.5 ]

             1.7    tan (180° − θ)                                [ 1.7 ]

             1.9    cos (360° − θ)                              [ 1.9 ]

           1.11    cot (360° − θ)                             [ 1.11 ]

           1.13    sin (90° − θ)                               [ 1.13 ]

           1.15    tan (270° − θ)                             [ 1.15 ]

           1.17    cosec (90° − θ)                         [ 1.17 ]

           1.19    tan (90° + θ)                              [ 1.19 ]

           1.21    sin (270° − θ)                            [ 1.21 ]

           1.23    sin (90° + θ)                              [ 1.23 ]

                 Write the following as ratios of θ alone :

             1.2    cos (180° − θ)                            [ 1.2 ]

             1.4    sec (360° + θ)                            [ 1.4 ]

             1.6    cot (180° + θ)                             [ 1.6 ]

             1.8    sin (180° + θ)                             [ 1.8 ]

           1.10    cosec (180° − θ)                     [ 1.10 ]

           1.12    tan (180° + θ)                          [ 1.12 ]

           1.14    cos (90° + θ)                           [ 1.14 ]

           1.16    cot (270° + θ)                         [ 1.16 ]

           1.18    sec (90° + θ)                          [ 1.18 ]

           1.20    sec (270° + θ)                        [ 1.20 ]

           1.22    cos (270° + θ)                        [ 1.22 ]

           1.24    cos (90° − θ)                          [ 1.24 ]

Vraag / Question 2

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

             2.1    sin 70°                                             [ 2.1 ]

             2.3    cosec 200°                                     [ 2.3 ]

             2.5    cot 250°                                          [ 2.5 ]

             2.7    sin 200°                                          [ 2.7 ]

             2.9    tan 290°                                         [ 2.9 ]

           2.11    cot 110°                                       [ 2.11 ]

           2.13    cot 200°                                       [ 2.13 ]

           2.15    sin 110°                                       [ 2.15 ]

           2.17    tan 110°                                       [ 2.17 ]

                 Write each of the following as a ratio of 20° :


             2.2    cos 110°                                     [ 2.2 ]

             2.4    sec 340°                                     [ 2.4 ]

             2.6    tan 380°                                      [ 2.6 ]

             2.8    cos 160°                                     [ 2.8 ]

           2.10    cosec 70°                                 [ 2.10 ]

           2.12    sec 250°                                   [ 2.12 ]

           2.14    cosec 250°                              [ 2.14 ]

           2.16    cos 70°                                    [ 2.16 ]

           2.18    sec 70°                                    [ 2.18 ]

Vraag / Question 3

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

             3.1    cos 50°                                           [ 3.1 ]

             3.3    cot 310°                                         [ 3.3 ]

             3.5    sin 230°                                         [ 3.5 ]

             3.7    cos 310°                                       [ 3.7 ]

             3.9    cosec 130°                                   [ 3.9 ]

           3.11    cos 130°                                     [ 3.11 ]

           3.13    cot 50°                                        [ 3.13 ]

           3.15    sec 140°                                     [ 3.15 ]

           3.17    sin 320°                                      [ 3.17 ]

                 Write each of the following as a ratio of 40° :


             3.2    cosec 320°                                 [ 3.2 ]

             3.4    tan 130°                                      [ 3.4 ]

             3.6    sec 400°                                     [ 3.6 ]

             3.8    sec 230°                                     [ 3.8 ]

           3.10    sin 220°                                    [ 3.10 ]

           3.12    tan 230°                                    [ 3.12 ]

           3.14    cosec 310°                              [ 3.14 ]

           3.16    cos 230°                                  [ 3.16 ]

           3.18    tan 140°                                   [ 3.18 ]

Vraag / Question 4

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

             4.1    sin 155°                                           [ 4.1 ]

             4.3    tan 145°                                           [ 4.3 ]

             4.5    sec 298°                                         [ 4.5 ]

             4.7    sin 138°                                          [ 4.7 ]

             4.9    cot 118°                                          [ 4.9 ]

           4.11    cot 288°                                        [ 4.11 ]

           4.13    cos 308°                                       [ 4.13 ]

           4.15    sec 218°                                       [ 4.15 ]

           4.17    cosec 232°                                   [ 4.17 ]

                 Write each of the following as a ratio
                 of an acute angle :

             4.2    cosec 222°                                 [ 4.2 ]

             4.4    cot 213°                                      [ 4.4 ]

             4.6    cosec 124°                                 [ 4.6 ]

             4.8    sec 318°                                     [ 4.8 ]

           4.10    cosec 346°                              [ 4.10 ]

           4.12    tan 113°                                   [ 4.12 ]

           4.14    sin 150°                                    [ 4.14 ]

           4.16    cot 128°                                   [ 4.16 ]

           4.18    cos 142°                                  [ 4.18 ]

Vraag / Question 5

                 As sin 23° = p, skryf die volgende
                 in terme van p :

             5.1    cos 67°                                           [ 5.1 ]

             5.3    cos 293°                                         [ 5.3 ]

             5.5    cos 113°                                         [ 5.5 ]

             5.7    cos 23°                                           [ 5.7 ]

             5.9    cosec 23°                                       [ 5.9 ]

                 If sin 23° = p, write the following
                 in terms of p :

             5.2    sin 157°                                       [ 5.2 ]

             5.4    sin 203°                                       [ 5.4 ]

             5.6    sin 337°                                       [ 5.6 ]

             5.8    cos 247°                                      [ 5.8 ]

           5.10    tan 23°                                       [ 5.10 ]

Vraag / Question 6

                 Vereenvoudig die volgende so ver as moontlik :    /    Simplify the following as far as posssible :


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


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


             6.3  tan (180° + x) ─ tan (180° ─ x) + tan (360° + x) ─ tan (360° ─ x)                                                 [ 6.3 ]


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


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


                        sin (360° ─ x) . tan (180° + x)
             6.6   ─────────────────────
                        cos x . cot (90° ─ x) . sec (180° + x)                                                                                         [ 6.6 ]


                        cos (180° + θ) . sin (180° ─ θ)
             6.7   ─────────────────────
                        cos (360° ─ θ) . tan (360° ─ θ)                                                                                                 [ 6.7 ]


                        sin (360° + x) ÷ cos (90° ─ x) ÷ sin (180° ─ x)
             6.8   ─────────────────────
                        tan (180° + x) ÷ tan (360° + x) ÷ cos (270° ─ x)                                                                       [ 6.8 ]


                                  cos² (180° ─ x)
             6.9   ─────────────────────
                        sin (360° ─ x) . sin (90° ─ x)                                                                                                    [ 6.9 ]


                           cos (90° + α) . sin ( ─ α)
           6.10   ─────────────────────
                        sin (180° + α) . tan (360° ─ α)                                                                                               [ 6.10 ]


                           cos ( ─ β) . sin (180° + β)
           6.11   ─────────────────────
                        tan (β ─ 180°) . sin (β + 90°)                                                                                                 [ 6.11 ]


                               sin ( ─ α) . sec ( ─ α) . tan (180° ─ α)
           6.12   ───────────────────────────────
                        sec (360° ─ α) . cos  (270° ─ α) . tan (180° + α)                                                                  [ 6.12 ]

Vraag / Question 7

                 Vereenvoudig die volgende sonder die gebruik van 'n sakrekenaar :    /

                 Simplify the following without the use of a pocket calculator :


             7.1  sin 320° . tan 140° . cot 220°. sec 130°                                                                                        [ 7.1 ]


             7.2  sin 200° . sec 110°  +  cos 160° . cosec 110°;                                                                              [ 7.2 ]


             7.3  cos 290° . tan 200°  ─  cos 160°                                                                                                   [ 7.3 ]


             7.4  cos 63° . cosec 207° . cot 27°                                                                                                       [ 7.4 ]


             7.5  sin 125° . cot 145° . cosec 55° . sin 215°                                                                                      [ 7.5 ]


                       cosec 110°
             7.6  ─────────
                       2 sec 20°                                                                                                                                    [ 7.6 ]


                       tan 340° . sin 72°
             7.7  ────────────
                       cos 18° . cot 70°                                                                                                                        [ 7.7 ]


                       sec 185° . sin 78°
             7.8  ────────────
                       cos 348° . sec 175°                                                                                                                   [ 7.8 ]


                       3 tan 65° . cosec 160°
             7.9  ────────────────
                       4 cosec 200° . cot 25°                                                                                                             [ 7.9 ]


                                          cot 27° . tan 153°
           7.10    ────────────────────────────
                       sin 27° . cos 117°  ─  cosec 207° . sin 333°                                                                           [ 7.10 ]


                            1
           7.11    ────── X cosec 128°  +  sec 232° . cos² 142°
                        cot 52°                                                                                                                                   [ 7.11 ]


                        sin 108° . tan 72° . sin 162°
           7.12    ────────────────────────────
                        tan 252° . cos ( ─ 72°)  .  sin ( ─ 72°)                                                                                   [ 7.12 ]

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