WISKUNDE
GRAAD 12
NOG OEFENINGE
Vergelykings - sin = sin, ens. : antwoorde.
MATHEMATICS
GRADE 12
MORE EXERCISES
Equations - sin = sin, etc. : answers.
1.1
sin (6θ − 10°) = sin (5θ + 2°)
6θ − 10° = 5θ + 2° + k . 360°
OF / OR
6θ − 10° = 180° − (5θ + 2°) + k . 360° ; k ∈ Z
θ = 12° + k . 360° ; k ∈ Z
6θ − 10° = 180° − 5θ − 2° + k . 360° ; k ∈ Z
11θ = 188° + k . 360° ; k ∈ Z
θ = 17,09° + k . 32,73° ; k ∈ Z
1.2
sin (2x − 3°) = sin (x + 19°)
2x − 3° = x + 19° + k . 360°
OF / OR
2x − 3° = 180° − (x + 19°) + k . 360° ; k ∈ Z
x = 22° + k . 360° ; k ∈ Z
3x = 164° + k . 360° ; k ∈ Z
x = 54,67° + k . 120° ; k ∈ Z
1.3
sin (4θ − 6°) = sin (3θ + 11°)
4θ − 6° = 3θ + 11° + k . 360°
OF / OR
4θ − 6° = 180° − (3θ + 11°) + k . 360° ; k ∈ Z
θ = 17° + k . 360° ; k ∈ Z
7θ = 175° + k . 360° ; k ∈ Z
θ = 25° + k . 51,43° ; k ∈ Z
1.4
cos (2x − 7°) = cos (x + 24°)
2x − 7° = x + 24° + k . 360°
OF / OR
2x − 7° = − (x + 24°)+ k . 360° ; k ∈ Z
x = 31° + k . 360° ; k ∈ Z
3x = − 17° + k . 360° ; k ∈ Z
x = − 5,67° + k . 120° ; k ∈ Z
1.5
cos (5α − 21°) = cos (4α + 3°)
5α − 21° = 4α + 3° + k . 360°
OF / OR
5α − 21° = − (4α + 3°)+ k . 360° ; k ∈ Z
α = 24° + k . 360° ; k ∈ Z
9α = 18° + k . 360° ; k ∈ Z
α = 2° + k . 40° ; k ∈ Z
1.6
cos (4x + 29°) = cos (x − 16°)
4x + 29° = x − 16° + k . 360°
OF / OR
4x + 29° = − (x − 16°)+ k . 360° ; k ∈ Z
3x = − 45° + k . 360° ; k ∈ Z
5x = − 13° + k . 120° ; k ∈ Z
x = − 15° + k . 120° ; k ∈ Z
x = − 2,6° + k . 72° ; k ∈ Z
1.7
tan (α − 6°) = tan (3α − 44°)
α − 6° = 3α − 44° + k . 360°
OF / OR
α − 6° = 180° + (3α − 44°)+ k . 360° ; k ∈ Z
2α = 38° + k . 360° ; k ∈ Z
2α = − 142° + k . 360° ; k ∈ Z
α = 19° + k . 180° ; k ∈ Z
α = − 71° + k . 180° ; k ∈ Z
1.8
tan (5θ − 18°) = tan (3θ + 8°)
5θ − 18° = 3θ + 8° + k . 360°
OF / OR
5θ − 18° = 180° + (3θ + 8°)+ k . 360° ; k ∈ Z
2θ = 26° + k . 360° ; k ∈ Z
2θ = 206° + k . 360° ; k ∈ Z
θ = 13° + k . 180° ; k ∈ Z
θ = 103° + k . 180° ; k ∈ Z
1.9
cos (2x + 20°) = − cos (x − 11°)
cos (2x + 20°) = cos [180° − (x − 11°)]
2x + 20° = 180° − x + 11°
OF / OR
2x + 20° = − (180° − x + 11° )+ k . 360° ; k ∈ Z
3x = 171° + k . 360° ; k ∈ Z
x = − 211° + k . 360° ; k ∈ Z
x = 57° + k . 120° ; k ∈ Z
1.10
sin (5x − 10°) = sin (2x + 50°)
5x − 10° = 2x + 50°
OF / OR
5x − 10° = 180° − (2x + 50°) + k . 360° ; k ∈ Z
3x = 60° + k . 360° ; k ∈ Z
7x = 140° + k . 360° ; k ∈ Z
x = 20° + k . 120° ; k ∈ Z
x = 20° + k . (180/7°) ; k ∈ Z
1.11
tan (7α + 20°) = − tan (3α − 40°)
7α + 20° = 180° − (3α − 40°)
OF / OR
7α + 20° = 180° + (180° − (3α − 40°)) + k . 360° ; k ∈ Z
10α = 200° + k . 360° ; k ∈ Z
10α = 380° + k . 360° ; k ∈ Z
α = 20° + k . 36° ; k ∈ Z
α = 38° + k . 36°) ; k ∈ Z
1.12
sin (3θ − 54°) = − sin (θ + 18°)
3θ − 54° = 180° + (θ + 18°)
OF / OR
3θ − 54° = 180° − (180° + (θ + 18°)) + k . 360° ; k ∈ Z
= 198° + θ + k . 360° ; k ∈ Z
= − θ − 18° + k . 360° ; k ∈ Z
2θ = 252° + k . 360° ; k ∈ Z
4θ = 36° + k . 360° ; k ∈ Z
θ = 126° + k . 180° ; k ∈ Z
θ = 9° + k . 90° ; k ∈ Z