WISKUNDE
GRAAD 12
NOG OEFENINGE
Vergelykings - funksiewaarde gegee : antwoorde.
MATHEMATICS
GRADE 12
MORE EXERCISES
Equations - function value given : answers.
1.1
sin (θ + 15°) = 0,588
Skerphoek / Acute angle = 36,01521°
sin (θ + 15°) is positief en dus is die
sin (θ + 15°) is positive and therefore the
hoek (θ + 15°) in die eerste en tweede kwadrante.
angle (θ + 15°) is in the first and second quadrants.
θ + 15° = 36,01521°
OF / OR
θ + 15° = 180° − 36,01521°
θ = 36,01521° − 15°
θ = (180° − 36,01521°) − 15°
= 21,021°
= 128,98°
1.2
cos 2x = − 0,616
Skerphoek / Acute angle = 51,97538...°
cos 2x is negatief en dus is die
cos 2x is negative and therefore the
hoek 2x in die tweede of derde kwadrante.
angle 2x is in the second or third quadrants.
2x = 180° − 51,97538...°
OF / OR
2x = 180° + 51,97538...°
= 128,02462...°
= 231,97538...°
x = 64,01°
x = 115,99°
1.3
tan 3θ = 1
Skerphoek / Acute angle = 45°
tan 3x > 0 en dus is
tan 3x > 0 and therefore
hoek 3x in die eerste of derde kwadrante.
angle 3x is in the first or third quadrants.
3x = 45°
OF / OR
3x = 180° + 45°
= 225°
x = 15°
x = 75°
1.4
sin (2x + 10°) = − 0,74
Skerphoek / Acute angle = 47,731415....°
sin (2x + 10°) < 0 en dus is
sin (2x + 10°) < 0 and therefore
die hoek in die derde of vierde kwadrante.
the angle is in the third or fourth quadrants.
(2x + 10°) = 180° + 47,731415....°
OF / OR
(2x + 10°) = 360° − 47,731415....°
= 227,731415...°
= 312,8585...°
2x = 217,731415..°
2x = 302,268585...°
x = 108,87°
x = 151,13°
1.5
2 cos (3x + 16°) = 0,684
cos (3x + 16°) = 0,342
Skerphoek / Acute angle = 70°
cos (3x + 16°) > 0 en dus is die hoek
cos (3x + 16°) > 0 and therefore the angle
in die eerste of vierde kwadrante.
is in the first or fourth quadrants.
3x + 16° = 70°
OF / OR
3x + 16° = 360° − 70°
= 290°
3x = 54°
3x = 274°
x = 18°
x = 91,33°
1.6
2 sin (3x + 1°) = 1,846
1,846
sin (3x + 1°) = ───── = 0,923
2
Skerphoek / Acute angle = 67,36867°
sin (3x + 1°) > 0 en dus is die hoek
sin (3x + 1°) > 0 and therefore the angle
in die eerste of tweede kwadrante.
is in the first or second quadrants.
3x + 1° = 67,36867...°
OF / OR
3x + 1° = 180° − 67,36867...°
= 112,63133...°
3x = 66,36867...°
3x = 111,63133...°
x = 22,12°
x = 37,21°
2.1
2 sin (2x − 21°) = 1,1
sin (2x − 21°) = 0,55
Skerphoek / Acute angle = 33,3670...°
sin (2x − 21°) > 0 en dus is die hoek
sin (2x − 21°) > 0 and therefore the angle
in die eerste of tweede kwadrante.
is in the first or second quadrants.
2x − 21° = 33,3670...° + k.360°; k∈Z
OF / OR
2x − 21° = 180° − 33,3670...° + k.360°; k∈Z
2x = 54,3670...° + k.360°
2x = 167,6330...° + k.360°
x = 27,18° + k.180°; k ∈ Z
x = 83,82° + k.180°; k ∈ Z
2.2
2 tan (3x + 18°) = 7
tan (3x + 18°) = 3,5
Skerphoek / Acute angle = 74,0546...°
tan (3x + 18°) > 0 en dus is die hoek
tan (3x + 18°) > 0 and therefore the angle
in die eerste of derde kwadrante.
is in the first or third quadrants.
3x + 18° = 74,0546...° + k.360°; k ∈ Z
3x = 56,0546...° + k.360°
x = 18,68° + k.60°; k ∈ Z
2.3
3 cos (2x + 34°) + 0,966 = 0
cos (2x + 34°) = − 0,322
Skerphoek / Acute angle = 71,2160...°
cos (2x + 34°) < 0 en dus is die hoek
cos (2x + 34°) < 0 and therefore the angle
in die tweede of derde kwadrante.
is in the second or third quadrants.
2x + 34° = 180° − 71,2160......° + k . 360°; k ∈ Z
2x = 74,784...° + k . 360°
x = 37,39° + k . 180°; k ∈ Z
OF / OR
2x + 34° = 180° + 71,2160......° + k . 360°; k ∈ Z
2x = 217,2160...° + k . 360°; k ∈ Z
x = 108,61° + k . 180°; k ∈ Z
2.4
4 sin (5x − 33°) + 0,832 = 0
sin (5x − 33°°) = − 0,208
Skerphoek / Acute angle = 12,0051...°
sin (5x − 33°) < 0 en dus is die hoek
sin (5x − 33°) < 0 and therefore the angle
in die derde of vierde kwadrante.
is in the third or fourth quadrants.
5x − 33° = 180° + 12,0051....° + k . 360°; k ∈ Z
5x = 225,0051...° + k . 360°; k ∈ Z
x = 45° + k . 72°; k ∈ Z
OF / OR
5x − 33° = 360° − 12,0051....° + k . 360°; k ∈ Z
5x = 380,9949...° + k . 360°; k ∈ Z
x = 76,20° + k . 72°; k ∈ Z
2.5
sin (2α − 40°) . cos (α + 15°) + cos (2α − 40°) . sin (α + 15°) = 0,78
[NB: sin A.cos b + cos A.sin B = sin (a + B)]
sin [(2α − 40°) + (α + 15°)] = 0,78
sin (3α − 25°) = 0,78
Skerphoek / Acute angle = 51,26°
3α − 25° = 51,26° + k . 360° ; k ∈ Z
OF / OR
3α − 25° = 180° − 51,26° + k . 360° ; k ∈ Z
3α = 76,26° + k . 360° ; k ∈ Z
3α = 153,74° + k . 360° ; k ∈ Z
α = 25,42° + k . 120° ; k ∈ Z
α = 51,25° + k . 120° ; k ∈ Z
2.6
sin (3θ + 10°) . cos (2θ − 31°) + cos (3θ + 10°) . sin (2θ − 31°) = 0,86
sin [(3θ + 10°) + (2θ − 31°)] = 0,86
sin (5θ − 21°) = 0,86
Skerphoek / Acute angle = 59,31658...°
5θ − 21° = 59,31658...° + k . 360°
OF / OR
5θ − 21° = 180° − 59,31658...° + k . 360° ; k ∈ Z
5θ = 80,31658...° + k . 360° ; k ∈ Z
5θ = 120,6834171...° + k . 360° ; k ∈ Z
θ = 16,06° + k . 72° ; k ∈ Z
θ = 24,14° + k . 72° ; k ∈ Z
2.7
sin (5α + 28°) . cos (3α + 41°) − cos (5α + 28°) . sin (3α + 41°) = 0,6
sin [(5α + 28°) − (3α + 41°)] = 0,6
sin (5α + 28°) = 0,6
Skerphoek / Acute angle = 36,8698...°
2α − 13° = 36,8698...° + k . 360°
OF / OR
2α − 13° = 180° − 36,8698...° + k . 360° ; k ∈ Z
2α = 49,8698...° + k . 360° ; k ∈ Z
2α = 156,1302...° + k . 360° ; k ∈ Z
α = 24,93° + k . 180° ; k ∈ Z
α = 78,07° + k . 180° ; k ∈ Z
2.8
sin (7x + 33°) . cos (3x + 21°) − 0,951 = cos (7x + 33°) . sin (3x + 21°)
sin [(7x + 33°) − (3x + 21°)] = 0,951
sin (4x + 12°) = 0,951
Skerphoek / Acute angle = 71,98952...°
4x + 12° = 71,98952...° + k . 360°
OF / OR
4x + 12° = 180° − 71,98952...° + k . 360° ; k ∈ Z
4x = 59,98952...° + k . 360° ; k ∈ Z
4x = 96,0104...° + k . 360° ; k ∈ Z
x = 15° + k . 90° ; k ∈ Z
x = 24° + k . 90° ; k ∈ Z
2.9
cos (2α − 19°) . cos (3α − 61°) − sin (2α − 19°) . sin (3α − 61°) = 0.766
cos [(2α − 19°) + (3α − 61°)] = 0,766
cos (5α − 80°) = 0,766
Skerphoek / Acute angle = 40°
5α − 80° = 40° + k . 360°
OF / OR
5α − 80° = − 40° + k . 360° ; k ∈ Z
5α = 120° + k . 360° ; k ∈ Z
5α = 40° + k . 360° ; k ∈ Z
α = 24° + k . 72° ; k ∈ Z
α = 8° + k . 72° ; k ∈ Z
2.10
cos (3θ − 41°) . cos (θ + 7°) − sin (3θ − 41°) . sin (θ + 7°) = 0.743
cos [(3θ − 41°) + (θ + 7°)] = 0,743
cos (4θ − 34°) = 0,743
Skerphoek / Acute angle = 42,012399...°
4θ − 34° = 42,012399...° + k . 360°
OF / OR
4θ − 80° = − 42,012399...° + k . 360° ; k ∈ Z
4θ = 76,012399...° + k . 360° ; k ∈ Z
4θ = − 8,012399...° + k . 360° ; k ∈ Z
θ = 19° + k . 90° ; k ∈ Z
θ = − 2° + k . 90° ; k ∈ Z
2.11
cos (5x + 18°) . cos (3x + 67°) + sin (5x + 18°) . sin (3x + 67°) = tan 16,3°
cos [(5x + 18°) − (3x + 67°)] = 0,29242..
cos (2x − 49°) = 0,29242..
Skerphoek / Acute angle = 88,3243...°
2x − 49° = 88,3243...° + k . 360°
OF / OR
2x − 49° = − 88,3243...° + k . 360° ; k ∈ Z
2x = 137,32431...° + k . 360° ; k ∈ Z
2x = − 39,3243...° + k . 360° ; k ∈ Z
x = 68,66° + k . 180° ; k ∈ Z
x = − 19,66° + k . 180° ; k ∈ Z
2.12
cos (7α + 36°) . cos (4α − 9°) + sin (7α + 36°) . sin (4α − 9°) = sin 12°
cos [(7α + 36°) − (4α − 9°)] = 0,207911...
cos (3α + 45°) = 0,207911...
Skerphoek / Acute angle = 77,99482...°
3α + 45° = 77,99482...° + k . 360°
OF / OR
3α + 45° = − 77,99482...° + k . 360° ; k ∈ Z
3α = 32,9948...° + k . 360° ; k ∈ Z
3α = − 122,994827...° + k . 360° ; k ∈ Z
α = 11° + k . 120° ; k ∈ Z
α = − 41° + k . 120° ; k ∈ Z
2.13
2 sin (2x + 15°) . cos (2x + 15°) = 0,956
NB : 2 sin A cos A = k dan / then sin 2A = k
sin (2 (2x + 15°)) = 0,956
Skerphoek / Acute angle = 72,94037...°
4x + 30° = 72,94037...° + k . 360°
OF / OR
4x + 30° = 180° − 72,94037...° + k . 360° ; k ∈ Z
4x = 42,94037...° + k . 360° ; k ∈ Z
4x = 77,0596...° + k . 360° ; k ∈ Z
x = 10,74° + k . 90° ; k ∈ Z
x = 19,26° + k . 90° ; k ∈ Z
2.14
2 sin (3θ − 38°) . cos (3θ − 38°) = 0,618
sin (2 (3θ − 38°)) = 0,618
Skerphoek / Acute angle = 38,17023...°
6θ − 76° = 38,17023...° + k . 360°
OF / OR
6θ − 76° = 180° − 38,17023...° + k . 360° ; k ∈ Z
6θ = 114,17...° + k . 360° ; k ∈ Z
6θ = 217,83...° + k . 360° ; k ∈ Z
θ = 19,03° + k . 60° ; k ∈ Z
θ = 36,31° + k . 60° ; k ∈ Z
2.15
2 cos2 (5α + 18°) − 1 = 0,208
NB : 2 cos2 A − 1 = k, dan / then cos 2A = k
cos (2 (5α + 18°)) = 0,208
Skerphoek / Acute angle = 77,9948...°
2(5α + 18°) = 77,9948...° + k . 360°
OF / OR
2(5α + 18°) = − 77,9948...° + k . 360° ; k ∈ Z
10α + 36° = 77,9948,17...° + k . 360° ; k ∈ Z
10α + 36° = − 77,9948...° + k . 360° ; k ∈ Z
10α = 41,9948...° + k . 360° ; k ∈ Z
10α = − 113,9948...° + k . 360° ; k ∈ Z
α = 4,2° + k . 60° ; k ∈ Z
α = − 11,40° + k . 60° ; k ∈ Z
2.16
2 cos2 (3x − 28°) = 1,371
2 cos2 (3x − 28°) − 1 = 0,371
cos (2 (3x − 28°)) = 0,371
Skerphoek / Acute angle = 68,2226...°
2(3x − 28°) = 68,2226...° + k . 360°
OF / OR
2(3x − 28°) = − 68,2226...° + k . 360° ; k ∈ Z
6x − 56° = 68,2226...° + k . 360° ; k ∈ Z
6x − 56° = − 68,2226...° + k . 360° ; k ∈ Z
6x = 124,2226...° + k . 360° ; k ∈ Z
6x = − 12,2226...° + k . 360° ; k ∈ Z
x = 20,70° + k . 60° ; k ∈ Z
x = − 2,04° + k . 60° ; k ∈ Z