WISKUNDE
GRAAD 12
NOG OEFENINGE
Identiteite : antwoorde.
MATHEMATICS
GRADE 12
MORE EXERCISES
Identities : answers.
cos 175° ⋅ sin 78°
sin 33° ⋅ tan 115°
LK / LHS = ─────────────
LK / LHS = ─────────────
cos 348° ⋅ cos 185°
cos 123° ⋅ tan 295°
cos (180° − 5°) ⋅ sin (90° − 12°)
sin 33° ⋅ tan 115°
= ─────────────────────
= ──────────────────────
cos (360° − 12°) ⋅ cos (185° + 5°)
cos (90° + 33°) ⋅ tan (180° + 115°)
− cos 5° ⋅ cos 12°
sin 33° ⋅ tan 115°
= ─────────────
= ─────────────
cos 12° ⋅ (− cos 5°)
(− sin 33°) ⋅ tan 115°
= 1
= − 1
= RK / RHS
= RK / RHS
3 tan 65° ⋅ sin (180° + 20°) ⋅ cos (360° − 25°)
3.
LK / LHS = ──────────────────────────────
4 sin (180° − 20°) ⋅ cos 25° ⋅ sin (90° + 25°)
3 tan 65° ⋅ (− sin 20°) ⋅ cos 25°
= ─────────────────────
4 sin 20° ⋅ cos 25° ⋅ cos 25°
3 × tan 65° × cos 25°
= − ──────────────
4 × cos 25° × cos 25°
3 × sin (90° −25°) × cos 25°
3 × cos 25° × cos 25°
= − ───────────────────────── = − ─────────────────────
4 × cos (90° −25°) × sin 25° × cos 25°
4 × sin 25° × cos 25° × cos 25°
3
= − ───────
4 sin 25°
tan 480° ⋅ sin 300° ⋅ cos 14° ⋅ sin (−135°)
4.
LK / LHS = ──────────────────────────────
sin 104° ⋅ cos 225°
tan (360° + 120°) ⋅ sin(360° − 60°) ⋅ cos 14° ⋅ (− sin(180° − 45°))
= ───────────────────────────────────────────
sin (90° + 14°) ⋅ cos (180° + 45°)
tan (180° − 60°) ⋅ (− sin 60°) ⋅ cos 14° ⋅ (− sin 45°)
= ──&────────────────────────────────
cos 14° ⋅ (− cos 45°)
√3
1
(− tan 60°) × (− ──) × (− ──)
2
√2
= ──────────────────────
1
− ───
√2
√3
√3
= ── × ───
1
2
3
= ──
2
= RK / RHS
cos (270° + α) ⋅ tan (180° + α)
5.
LK / LHS = ────────────────────
sin (90° − α)
sin α ⋅ tan α
= ───────── = tan α ⋅ tan α
cos α
= tan2 α
= RK / RHS
sin (180° − θ) ⋅ sin (90° − θ) ⋅ tan (180° − θ) ⋅ cos (− θ)
6.
LK / LHS = ────────────────────────────────────
cos (90° + θ) ⋅ cos (θ − 360°) ⋅ tan (−θ)
sin θ ⋅ cos θ ⋅ tan θ ⋅ cos θ
= ─────────────────────────
(− sin θ) ⋅ cos (−(360° − θ)) ⋅ (− tan θ)
sin θ ⋅ cos θ ⋅ tan θ ⋅ cos θ
= ───────────────────
sin θ ⋅ cos θ ⋅ tan θ
= cos θ
= RK / RHS
1
1
7.
LK / LHS = ───────────── − ────────────
sin (180° − θ) − 1
cos (90° − θ) + 1
1
1
= ──────── − ────────
sin θ − 1
sin θ + 1
1(sin θ + 1) − 1(sin θ − 1)
= ──────────────────
(sin θ + 1)(sin θ − 1)
sin θ + 1 − sin θ + 1
= ──────────────
sin2 θ − 1
2
= ───────
cos2 θ
= RK / RHS
tan (180° + x) ⋅ cos (360° − x)
8.
LK / LHS = ─────────────────────────────────────
sin (180° − x) ⋅ cos (90° + x) + cos (540° + x) ⋅ cos (− x)
tan x ⋅ cos x
tan x ⋅ cos x
= ─────────────────────────── = ───────────────────────
sin x ⋅ (− sin x) + cos (180° + x) ⋅ cos x
sin x ⋅ (− sin x) + (− cos x) ⋅ cos x
tan x ⋅ cos x
tan x ⋅ cos x
= ───────────── = ─────────
−(sin2 x + cos x)
− 1
sin x × cos x
= − ───────────
cos x
= − sin x
= RK / RHS
sin 190° ⋅ cos 225° ⋅ tan 390°
9.
LK / LHS = ────────────────────
cos 100° ⋅ sin 135°
sin (180° + 10°) ⋅ cos (180° + 45°) ⋅ tan (360° + 30°)
= ───────────────────────────────────
cos (90° + 10°) ⋅ sin (90° + 45°)
(− sin 10°) ⋅ (− cos 45°) ⋅ tan 30°
= ────────────────────────
(− sin 10°) ⋅ cos 45°
= − tan 30°
1
= − ───
√3
= RK / RHS