WISKUNDE
GRAAD 12
NOG OEFENINGE
Berekeninge sonder sakrekenaars : antwoorde.
MATHEMATICS
GRADE 12
MORE EXERCISES
Calculations without using pocket calculators : answers.
sin 220° ⋅ tan 220°
─────────────────────
cos 540° ⋅ tan 140° ⋅ sin 320°
sin (180° + 40°) ⋅ tan (180° + 40°)
= ────────────────────────────────────
cos (360° + 180°) ⋅ tan (180° − 40°) ⋅ sin (360° − 40°)
(− sin 40°) ⋅ tan 40°
= ────────────────────────
cos 180° ⋅ (− tan 40°) ⋅ (− sin 40°)
sin 40° ⋅ tan 40°
= − ──────────────────
(− 1) ⋅ tan 40° ⋅ sin 40°
= 1
sin 210° ⋅ sin 300° ⋅ tan 30°
────────────────────
cos 120° ⋅ tan 300° ⋅ sin 330°
sin (180° + 30°) ⋅ sin (270° + 30°) ⋅ tan 30°
= ───────────────────────────────────
cos (180° − 60°) ⋅ tan (360° − 60°) ⋅ sin (360° − 30°)
(− sin 30°) ⋅ cos 30° ⋅ tan 30°
= ─────────────────────────
(− cos 60°) ⋅ (− tan 60°) ⋅ (− sin 30°)
(− sin 30°) ⋅ cos 30° ⋅ tan 30°
= ─────────────────────────
(− cos 60°) ⋅ (− tan 60°) ⋅ (− sin 30°)
sin 30°
sin 30° ⋅ cos 30° ⋅ ──────
cos 30°
= − ─────────────────────────
sin 60°
cos 60° ⋅ ────── ⋅ sin 30°
cos 60°
1
───
2
sin 30°
= − ─────── = ──────
sin 60°
√3
───
2
1
= − ────
√3
sin 150° ⋅ cos 210°
sin (180° − 30°) ⋅ cos (180° + 30°)
──────────────
= ───────────────────────
tan 330°
tan (360° − 30°)
sin 30° ⋅ (− cos 30°)
= ────────────────
(− tan 30°)
cos 30°
= sin 30° ⋅ cos 30° × ──────
sin 30°
(√3)2
= cos2 30° = ─────
(2)2
3
= ──
4
tan 225°
= tan (180° + 75°)
= tan 75°
sin 75°
= ──────
cos 75°
sin (30° + 45°)
= ───────────
cos (30° + 45°)
sin 30° cos 45° + sin 45° cos 30°
= ──────────────────────
cos 30° cos 45° − sin 30° sin 45°
1
1
1
√3
── × ─── + ─── × ──
2
√2
√2
2
= ──────────────────────
√3
1
1
1
─── × ─── − ── × ───
2
√2
2
√2
1
┏
1
√3
┓
── ┃ ── + ── ┃
√2
┗
2
2
┛
= ───────────────
1
┏
√3
1
┓
── ┃ ── − ── ┃
√2
┗
2
2
┛
√3
1
── + ──
2
2
= ─────────
√3
1
── − ──
2
2
√3 + 1
2
= ───── × ─────
2
√3 − 1
√3 + 1
√3 + 1
= ───── × ─────
. . . Rasionaliseer noemer / Rationalise denominator
√3 − 1
√3 + 1
(√3)2 + 2√3 + 1
4 + 2√3
= ─────────── = ───────
(√3)2 − 1
2
= 2 + √3
cos (−45°) ⋅ sin 45° − sin 210°
cos 45° ⋅ sin 45° − sin (180° + 30°)
─────────────────────
= ─────────────────────────
tan 315°
tan (360° − 45°)
cos 45° ⋅ sin 45° − (− sin 30°)
= ─────────────────────────
− tan 45°
1
1
1
1
1
── × ── + ──
── + ──
√2
√2
2
2
2
= ────────────
= ───────
− 1
− 1
= − 1
cos 55° ⋅ cos 35°
cos 55° ⋅ cos (90° − 55°)
──────────────
= ──────────────────
sin 110° ⋅ sin 270°
sin (2 × 55°) ⋅ sin 270°
cos 55° ⋅ sin 55°
= ──────────────────
2sin 55° cos 55° ⋅ (− 1)
1
= − ──
2
sin 210° ⋅ cos 300° ⋅ tan 240°
sin (180° + 30°) ⋅ cos (270° + 30°) ⋅ tan (180° + 60°)
─────────────────────
= ───────────────────────────────────
cos 120° ⋅ tan 150° ⋅ sin 330°
cos (90° + 30°) ⋅ tan (180° − 30°) ⋅ sin (360° − 30°)
(− sin 30°) ⋅ (− sin 30°) ⋅ tan 60°
= ────────────────────────
(− sin 30°) ⋅ (− tan 30°) ⋅ (− sin 30°)
tan 60°
= − ──────
tan 30°
√3
= − ──── = − √3 × √3
1
───
√3
= −3
sin (65° − x) ⋅
cos (35° − x) − cos (65° − x) ⋅ sin (35° − x)
= sin [(65° − x) − (35° − x)]
. . . sin (A − B) = sin A ⋅ cos B − cos A ⋅ sin B
= sin (65° − x − 35° + x)
= sin (30°
1
= ──
2
cos 43° ⋅ cos 17° − sin 43° ⋅ sin 17°
= cos (43° + 17°)
. . . cos (A + B) = cos A ⋅ cos B + sin A ⋅ sin B
= cos 60°
1
= ──
2
cos 15° = cos (45° − 30°)
= cos 45° ⋅ cos 30° + sin 45° ⋅ sin 30°
1
√3
1
1
√3 + 1
= ─── × ─── + ─── × ──
= ─────
√2
2
√2
2
2√2
√3 + 1
2√2
= ───── × ────
. . . rasionaliseer noemer / rationalise the denominator
2√2
2√2
2√6 + 2√2
= ────────
4 × 2
√6 + √2
= ────────
4
tan (− 225°)
= − tan 225°
= − tan (180° + 45°)
= − tan 45°
= −1
(sin 75° + cos 75°)2
= sin2 75° + 2sin 75° cos 75° + cos2 75°
. . . (a + b)2 = a2 + 2ab + b2
= 1 + 2sin 75° cos 75°
. . .sin2 x + cos2 x = 1
= 1 + sin 150°
. . .2sin x cos x = sin 2x
= 1 + sin (180° − 30°)
= 1 + sin 30°
1
= 1──
2