Pas die volgende reëls toe as jy die
formules gebruik :
1. Die teken van die funksie word bepaal deur
die oorspronklike funksie in die
oorspronklike kwadrant deur die CAST
reël te gebruik.
2. Wanneer die waarde van die funksies van
(180° ± θ) of (360° − θ) bepaal word, verander
die funksie nooit nie maar die
teken mag verander.
3. Wanneer die waarde van die funksies van
(90° ± θ) of (270° ± θ) bepaal word, verander
die funksie na sy ko-funksie en die
teken mag verander.
Apply the following rules when using the
reduction formulae :
1. The sign of the function is determined
from the original function in the
original quadrant by using the CAST rule.
2. When determining the value of the functions
of (180° ± θ) or (360° − θ),
the function never changes but
the sign may change.
3. When determining the value of the functions
of (90° ± θ) or (270° ± θ),
the function changes to its co-function and
the sign may change.
1.1 sin (180° − θ) = sin θ
┃
180° − θ
: naam onveranderd; hoek in kwadrant 2; sin (180° − θ) > 0 [CAST]
┃
180° − θ
: name unchanged; angle in quadrant 2; sin (180° − θ) > 0 [CAST]
[ 1.1 ]
1.2 cos (180° − θ) = ─ cos θ ┃
180° − θ
: name unchanged; angle in quadrant 2; cos (180° − θ) < 0 [CAST]
┃
180° − θ
: naam onveranderd; hoek in kwadrant 2; cos (180° − θ) < 0 [CAST]
[ 1.2 ]
1.3 tan (360° − θ) = ─ tan θ ┃
360° − θ
: naam onveranderd; hoek in kwadrant 4; tan (360° − θ) < 0 [CAST]
┃
360° − θ
: name unchanged; angle in quadrant 4; tan (360° − θ) < [CAST]
[ 1.3 ]
1.4 sec (360° + θ) = sec θ ┃
360° + θ
: naam onveranderd; hoek in kwadrant 1; sec (360° + θ) > 0 [CAST]
┃
360° + θ
: name unchanged; angle in quadrant 1; sec (360° + θ) > 0 [CAST]
[ 1.4 ]
1.5 cosec (360° − θ) = − cosec θ
┃
360° − θ
: naam onveranderd; hoek in kwadrant 4; cosec (360° + θ) < 0 [CAST]
┃
360° − θ
: name unchanged; angle in quadrant 4; cosec (360° + θ) < 0 [CAST]
[ 1.5 ]
1.6 cot (180° + θ) = cot θ ┃
180° + θ
: naam onveranderd; hoek in kwadrant 3; cot (180° + θ) > 0 [CAST]
┃
180° + θ
: name unchanged; angle in quadrant 3; cot (180° + θ) > 0 [CAST]
[ 1.6 ]
1.7 tan (180° − θ) = ─ tan θ
[ 1.7 ]
1.8 sin (180° + θ) = − sin θ
[ 1.8 ]
1.9 cos (360° − θ) = cos θ
[ 1.9 ]
1.10 cosec (180° − θ) = cosec θ
[ 1.10 ]
1.11 cot (360° − θ) = ─ cot θ
[ 1.11 ]
1.12 tan (180° + θ) = tan θ
[ 1.12 ]
1.13 sin (90° − θ) = cos θ ┃
90° − θ
: naam verander; hoek in kwadrant 1; cot (90° − θ) > 0 [CAST]
┃
90° − θ
: name changes; angle in quadrant 1; sin (90° − θ) > 0 [CAST]
[ 1.13 ]
1.14 cos (90° + θ) = − sin θ ┃
90° + θ
: naam verander; hoek in kwadrant 2; cos (90° + θ) < 0 [CAST]
┃
90° + θ
: name changes; angle in quadrant 2; cos (90° + θ) < 0 [CAST]
[ 1.14 ]
1.15 tan (270° − θ) = cot θ ┃
270° − θ
: naam verander; hoek in kwadrant 3; tan (270° − θ) > 0 [CAST]
┃
270° − θ
: name changes; angle in quadrant 3; tan (270° − θ) > 0 [CAST]
[ 1.15 ]
1.16 cot (270° + θ) = − tan θ ┃
270° + θ
: naam verander; hoek in kwadrant 4; cot (270° + θ) < 0 [CAST]
┃
270° + θ
: name changes; angle in quadrant 4; cot (270° + θ) < 0 [CAST]
[ 1.16 ]
1.17 cosec (90° − θ) = sec θ ┃
90° − θ
: naam verander; hoek in kwadrant 1; cosec (90° − θ) > 0 [CAST]
┃
90° − θ
: name changes; angle in quadrant 1; cosec (90° − θ) > 0 [CAST]
[ 1.17 ]
1.18 sec (90° + θ) = − cosec θ ┃
90° + θ
: naam verander; hoek in kwadrant 2; sec (90° + θ) < 0 [CAST]
┃
90° + θ
: name changes; angle in quadrant 2; sec (90° + θ) < 0 [CAST]
[ 1.18 ]
1.19 tan (90° + θ) = − cot θ
[ 1.19 ]
1.20 sec (270° + θ) = cosec θ
[ 1.20 ]
1.21 sin (270° − θ) = − cos θ
[ 1.21 ]
1.22 cos (270° + θ) = − sin θ
[ 1.22 ]
1.23 sin (90° + θ) = cos θ
[ 1.23 ]
1.24 cos (90° − θ) = sin θ
[ 1.24 ]
2.1 sin 70° = sin (90° − 20°)
= cos 20°
[ 2.1 ]
2.3 cosec 200° = cosec (180° + 20°)
= − cosec 20°
[ 2.3 ]
2.5 cot 250° = cot (270° − 20°)
= tan 20°
[ 2.5 ]
2.7 sin 200° = sin (180° + 20°)
= − sin 20°
[ 2.7 ]
2.9 tan 290° = tan (270° + 20°)
= − cot 20°
[ 2.9 ]
2.11 cot 110° = cot (90° + 20°)
= − tan 20°
[ 2.11 ]
2.13 cot 200° = cot (180° + 20°)
= cot 20°
[ 2.13 ]
2.15 sin 110° = c (90° + 20°)
= cos 20°
[ 2.15 ]
2.17 tan 110° = tan (90° + 20°)
= − cot 20°
[ 2.17 ]
2.2 cos 110° = cos (90° + 20°)
= − sin 20°
[ 2.2 ]
2.4 sec 340° = sec (360° − 20°)
= sec 20°
[ 2.4 ]
2.6 tan 380° = tan (360° + 20°)
= tan 20°
[ 2.6 ]
2.8 cos 160° = cos (180° − 20°)
= − cos 20°
[ 2.8 ]
2.10 cosec 70° = cosec (90° − 20°)
= sec 20°
[ 2.10 ]
2.12 sec 250° = sec (270° − 20°)
= − cosec 20°
[ 2.12 ]
2.14 cosec 250° = cosec (270°−20°)
= − sec 20°
[ 2.14 ]
2.16 cos 70° = cos (90° − 20°)
= sin 20°
[ 2.16 ]
2.18 sec 70° = sec (90° − 20°)
= cosec 20°
[ 2.18 ]
3.1 cos 50° = cos (90° − 40°)
= sin 40°
[ 3.1 ]
3.3 cot 310° = cot (270° + 40°)
= − tan 40°
[ 3.3 ]
3.5 sin 230° = sin (270° − 40°)
= − cos 40°
[ 3.5 ]
3.7 cos 310° = cos (270° + 40°)
= sin 40°
[ 3.7 ]
3.9 cosec 130° = cosec (90°−40°)
= sec 40°
[ 3.9 ]
3.11 cos 130° = cos (90° + 40°)
= − sin 40°
[ 3.11 ]
3.13 cot 50° = cot (90° − 40°)
= tan 40°
[ 3.13 ]
3.15 sec 140° = sec (180° − 40°)
= − sec 40°
[ 3.15 ]
3.17 sin 320° = sin (360° − 40°)
= − sin 40°
[ 3.17 ]
3.2 cosec 320° = cosec (360°−40°)
= − cosec 40°
[ 3.2 ]
3.4 tan 130° = tan (90° + 40°)
= − cot 40°
[ 3.4 ]
3.6 sec 400° = sec (360° + 40°)
= sec 40°
[ 3.6 ]
3.8 sec 230° = sec (270° − 40°)
= − cosec 40°
[ 3.8 ]
3.10 sin 220° = sin (180° + 40°)
= − sin 40°
[ 3.10 ]
3.12 tan 230° = tan (270° − 40°)
= cot 40°
[ 3.12 ]
3.14 cosec 310° = cosec (270°+40°)
= − sec 40°
[ 3.14 ]
3.16 cos 230° = cos (270° − 40°)
= − sin 40°
[ 3.16 ]
3.18 tan 140° = tan (180° − 40°)
= − tan 40°
[ 3.18 ]
4.1 sin 155° = sin (180° − 25°)
= sin 25°
OF / OR
sin 155° = sin (90° + 65°)
= cos 65°
[ 4.1 ]
4.3 tan 145° = tan (180° − 45°)
= − tan 45°
OF / OR
tan 145° = tan (90° + 55°)
= − cot 55°
[ 4.3 ]
4.5 sec 298° = sec (360° − 62°)
= sec 62°
OF / OR
sec 298° = sec (270° + 28°)
= cosec 28°
[ 4.5 ]
4.7 sin 138° = sin (180° − 42°)
= sin 42°
OF / OR
sin 138° = sin (90° + 48°)
= cos 48°
[ 4.7 ]
4.9 cot 118° = cot (180° − 62°)
= − cot 62°
[ 4.9 ]
4.11 cot 288° = cot (360° − 72°)
= − cot 72°
[ 4.11 ]
4.13 cos 308° = cos (360°−52°)
= cos 52°
[ 4.13 ]
4.15 sec 218° = sec (180° + 38°)
= − sec 38°
[ 4.15 ]
4.17 cosec 232° = cosec (180°+52°)
= − cosec 52°
[ 4.17 ]
4.2 cosec 222° = cosec (180°+42°)
= − cosec 42°
OF / OR
cosec 222° = cosec (270°−48°)
= − sec 48°
[ 4.2 ]
4.4 cot 213° = cot (180° + 33°)
= cot 33°
OF / OR
cot 213° = cot (270° − 57°)
= tan 57°
[ 4.4 ]
4.6 cosec 124° = cosec (180°−56°)
= cosec 56°
OF / OR
cosec 124° = cosec (90°+34°)
= sec 34°
[ 4.6 ]
4.8 sec 318° = sec (360° − 42°)
= sec 42°
OF / OR
sec 318° = sec (270° + 48°)
= cosec 48°
[ 4.8 ]
4.10 cosec 346° = cosec (360°−14°)
= − cosec 14°
[ 4.10 ]
4.12 tan 113° = tan (180° − 67°)
= − tan 67°
[ 4.12 ]
4.14 sin 150° = sin (180°−30°)
= sin 30°
[ 4.14 ]
4.16 cot 128° = cot (180° − 52°)
= − cot 52°
[ 4.16 ]
4.18 cos 142° = cos (180° − 38°)
= − cos 38°
[ 4.18 ]
Gebruik (180° ± 23°) OF (360° ± 23°) om die gegewe sin verhouding om te skakel na sin 23°
Gebruik (90° ± 23°) OF (270° ± 23°) om die gegewe cos verhouding om te skakel na sin 23°
Use (180° ± 23°) OR (360° ± 23°) to change the given sin ratio to sin 23°
Use (90° ± 23°) OR (270° ± 23°) to change the given cos ration to sin 23°
5.1 cos 67° = cos (90° − 23°)
= sin 23°
= p
[ 5.1 ]
5.3 cos 293° = cos (270° + 23°)
= sin 23
= p
[ 5.3 ]
5.5 cos 113° = cos (90° + 23°)
= − sin 23
= − p
[ 5.5 ]
5.7 sin
2 23° + cos
2 23° = 1
__________
cos 23° = √1 − sin
2 23°
______
= √1 − p
2
[ 5.7 ]
5.2 sin 157° = sin (180° − 23°)
= sin 23
= p
[ 5.2 ]
5.4 sin 203° = sin (180° + 23°)
= − sin 23°
= − p
[ 5.4 ]
5.6 sin 337° = sin (360° − 23°)
= − sin 23
= − p
[ 5.6 ]
5.8 cos 247° = cos (270° − 23°)
− sin 23
= − p
[ 5.8 ]
1
5.9 cosec 23° = ————
sin 23°
1
= ——
p
[ 5.9 ]
sin 23°
5.10 tan 23° = ————
cos 23°
p
= —————
______
√1 − p
2
______
p √1 − p
2
= ——————
1 − p
2
[ 5.10 ]
6.1 sec (180° + x)
. cosec (180° ─ x)
. cos (360° ─ x)
. sin (180° + x)
= ( ─sec x)
. (cosec x)
. (cos x)
. ( ─ sin x)
1 1
= ───── X ───── X cos x X sin x
cos x sin x
= 1
[ 6. ]
6.2 tan (180° + x)
. cos (360° ─ x)
. sec (90° + x)
= ( tan x)
. (cos x)
. ( ─ cosec x)
sin x 1
= ─ ───── X cos x X ─────
cos x sin x
= ─ 1
[ 6. ]
6.3 tan (180° + x)
─ tan (180° ─ x)
+ tan (360° + x)
─ tan (360° ─ x)
= ( tan x)
─ (─ tan x)
+ ( tan x) ─ (─ tan x)
= 4 tan x
[ 6. ]
6.4 sin (180° ─ x)
. cos (90° ─ x)
─ cos (180° + x)
. cos (360° ─ x)
= ( sin x)
. (sin x)
─ ( ─ cos x)
. (cos x)
= sin
2 x + cos
2 x
= 1
[ 6. ]
6.5 sin (90° + x)
. cos (90° ─ x)
+ cos (90° + x)
. sin (90° ─ x)
= ( cos x)
. (sin x)
+ ( ─ sin x)
. (cos x)
= 0
[ 6. ]
sin (360° ─ x)
. tan (180° + x)
6.6 ─────────────────────
cos x
. cot (90° ─ x)
. sec (180° + x)
( ─ sin x)
. (tan x)
= ─────────────────────
1
cos x
. (tan x)
. ( ─ ──── )
cos x
= sin x
[ 6. ]
cos (180° + θ)
. sin (180° ─ θ)
6.7 ─────────────────────
cos (360° ─ θ)
. tan (360° ─ θ)
( ─ cos θ)
. (sin θ)
= ──────────────
(cos θ)
. ( ─ tan θ)
sin θ cos θ
= ───── X ─────
1 sin θ
= cos θ
[ 6. ]
sin (360° + x)
÷ cos (90° ─ x)
÷ sin (180° ─ x)
6.8 ───────────────────────────────
tan (180° + x)
÷ tan (360° + x)
÷ cos (270° ─ x)
(sin x)
÷ (sin x)
÷ (sin x)
= ──────────────────
(tan x)
÷ (tan x)
÷ ( ─ sin x)
1
÷ (sin x)
= ─ ─────────
1
÷ (sin x)
= ─ 1
[ 6. ]
cos
2 (180° ─ x)
6.9 ─────────────────────
sin (360° ─ x)
. sin (90° ─ x)
( ─ cos x)
2
= ─────────────
( ─ sin x)
. (cos x)
cos x
= ─ ─────
sin x
= ─ cot x
[ 6. ]
cos (90° + α)
. sin ( ─ α)
6.10 ─────────────────────
sin (180° + α)
. tan (360° ─ α)
( ─ sin α)
. ( ─ sin α)
= ──────────────
( ─ sin α)
. ( ─ tan α)
sin α cos α
= ───── X ─────
1 sin α
= cos α
[ 6. ]
cos ( ─ β)
. sin (180° + β)
6.11 ─────────────────────
tan (β ─ 180°)
. sin (β + 90°)
(cos β)
. ( ─ sin β)
= ───────────────────
( ─ tan (180° ─ β))
. (cos β)
cos β
. sin β
= ───────────────────
( ─ tan β)
. cos β
sin β cos β
= ─ ───── X ─────
1 sin β
= ─ cos β
[ 6. ]
sin ( ─ α)
. sec ( ─ α)
. tan (180° ─ α)
6.12 ───────────────────────────────
sec (360° ─ α)
. cos (270° ─ α)
. tan (180° + α)
( ─ sin α)
. (sec α)
. ( ─ tan α)
= ─────────────────────
(sec α)
. ( ─ sin α)
. (tan α)
= ─ 1
[ 6. ]
7.1 sin 320°
. tan 140°
. cot 220°
. sec 130°
= sin (360° ─ 40°)
. tan (180° ─ 40°)
. cot (180° + 40°)
. sec (90° ─ 40°)
= ( ─ sin 40°)
. ( ─ tan 40°)
. (cot 40°)
. ( ─ cosec 40°)
sin 40° cos 40° 1
= ─ sin 40° X ───── X ────── X ─────
cos 40° sin 40° sin 40°
= ─ 1
[ 7. ]
7.2 sin 200°
. sec 110° + cos 160°
. cosec 110°;
= sin (180° + 20°)
. sec (90° + 20°) + cos (180° ─ 20°)
. cosec (90° + 20°)
= ( ─ sin 20°)
. ( ─ cosec 20°) + ( ─ cos 20°)
. (sec 20°)
= 1 + ( ─1)
= 0
[ 7. ]
7.3 cos 290°
. tan 200° ─ cos 160°
= cos (270° + 20°)
. tan (180° + 20°) ─ cos (180° ─ 20°)
= ( ─ sin 20°)
. ( ─ (tan 20°) ─ ( ─ cos 20°)
sin 20°
= sin 20° X ───── + cos 20°
cos 20°
sin
2 20° cos
2 20°
= ────── + ──────
cos 20° cos 20°
1
= ──────
cos 20°
= sec 20°
[ 7. ]
7.4 cos 63°
. cosec 207°
. cot 27°
= cos (90° ─ 27°)
. cosec (180° + 27°)
. cot 27°
= (sin 27°)
. ( ─ cosec 27°)
. (cot 27°)
= ─ cot 27°
[ 7. ]
7.5 sin 125°
. cot 145°
. cosec 55°
. sin 215°
= sin (90° + 35°)
. cot (180° ─ 35°)
. cosec (90° ─ 35°)
. sin (180° + 35°)
= cos 35°
. ( ─ cot 35°)
. sec 35°
. ( ─ sin 35°)
cos 35° 1
= cos 35° X ────── X ────── X sin 35°
sin 35° cos 35°
= cos 35°
[ 7. ]
cosec 110°
7.6 ─────────
2 sec 20°
cosec (90° + 20°)
= ────────────
2 sec 20°
sec 20°
= ────────
2 sec 20°
1
= ──
2
[ 7. ]
tan 340°
. sin 72°
tan (270° + 70°)
. sin 72°
7.7 ──────────── = ─────────────────
cos 18°
. cot 70°
cos (90° ─ 72°)
. cot 70°
( ─ cot 70°)
. sin 72°
= ─────────────────
(sin 72°)
. cot 70°
= ─ 1
[ 7. ]
sec 185°
. sin 78°
sec (180° + 5°)
. sin 78°
7.8 ──────────── = ───────────────────────
cos 348°
. sec 175°
cos (270° + 78°)
. sec (180° ─ 5°)
( ─ sec 5°)
. sin 78°
= ────────────────
(sin 78°)
. ( ─ sec 5°)
= 1
[ 7. ]
3 tan 65°
. cosec 160°
3 tan 65°
. cosec (180° ─ 20°)
7.9 ──────────────── = ─────────────────────────
4 cosec 200°
. cot 25°
4 cosec (180° + 20°)
. cot (90° ─ 65°)
3 (tan 65°)
. (cosec 20°)
= ────────────────
4 ( ─ cosec 20°)
. (tan 65°)
3
= ─ ──
4
[ 7. ]
cot 27°
. tan 153°
7.10 ────────────────────────────
sin 27°
. cos 117°
─ cosec 207°
. sin 333°
cot 27°
. tan (180° ─ 27°)
= ────────────────────────────────────────────
sin 27°
. cos (90° + 27°) ─ cosec (180° + 27°)
. sin (360° ─ 27°)
cot 27°
. ( ─ tan 27°)
= ─────────────────────────────────────
sin 27°
. ( ─ sin 27°) ─ ( ─ cosec 27°)
. ( ─ sin 27°)
─ 1
= ─────────────
─ sin
2 27° ─ 1
1
= ─────────────
sin
2 27° + 1
[ 7. ]
1
7.11 ────── X cosec 128° + sec 232°
. cos
2 142°
cot 52°
= tan 52°
. cosec (180° ─ 52°) + sec (180° ─ 52°)
. (cos (90° + 52°))
2
= tan 52°
. cosec 52° + ( ─ sec 52°)
. ( ─ sin 52°)
2
sin 52° 1 ( ─ 1 ) sin
2 52°
= ────── X ────── + ────── X ──────
cos 52° sin 52° cos 52° 1
1 sin
2 52°
= ────── ─ ──────
cos 52° cos 52°
1 ─ sin
2 52°
= ──────────
cos 52°
= cos 52°
[ 7. ]
sin 108°
. tan 72°
. sin 162°
7.12 ────────────────────────────
tan 252°
. cos ( ─ 72°)
. sin ( ─ 72°)
sin (180° ─ 72°)
. tan 72°
. sin (90° + 72°)
= ────────────────────────────
tan (180° + 72°)
. (cos 72°)
. ( ─ sin 72°)
sin 72°
. tan 72°
. cos 72°
= ─ ─────────────────────────
tan 72°
. cos 72°
. sin 72°
= ─ 1
[ 7. ]