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
reason 180° − θ : name unchanged;
angle in quadrant 2;
sin (180° − θ) > 0 . . . CAST
[ Q 1.1 ]
1.2 cos (180° − θ) = − cos θ
reason 180° − θ : name unchanged;
angle in quadrant 2;
cos (180° − θ) < 0 . . . CAST
[ Q 1.2 ]
1.3 tan (360° − θ) = − tan θ
reason
360° − θ : name unchanged;
angle in quadrant 4;
tan (360° − θ) < 0 . . . CAST
[ Q 1.3 ]
1.4 sec (360° + θ) = sec θ
reason
360° + θ : name unchanged;
angle in quadrant 1;
sec (360° + θ) > 0 . . . CAST
[ Q 1.4 ]
1.5 cosec (360° − θ) = − cosec θ
reason
360° − θ : name unchanged;
angle in quadrant 4;
cosec (360° − θ) < 0 . . . CAST
[ Q 1.4 ]
1.6 cot (180° + θ) = cot θ
reason
180° + θ : name unchanged;
angle in quadrant 3;
cot (180° + θ) > 0 . . . CAST
[ Q 1.5 ]
1.7 tan (180° − θ) = − tan θ
[ Q 1.7 ]
1.8 sin (180° + θ) = − sin θ
[ Q 1.8 ]
1.9 cos (360° − θ) = cos θ
[ Q 1.9 ]
1.10 cosec (180° − θ) = cosec θ
[ Q 1.10 ]
1.11 cot (360° − θ) = − cot θ
[ Q 1.11 ]
1.12 tan (180° + θ) = tan θ
[ Q 1.12 ]
1.13 sin (90° − θ) = cos θ
reason
90° − θ : name changes;
angle in quadrant 1;
sin (90° − θ) > 0 . . . CAST
[ Q 1.13 ]
1.14 cos (90° + θ) = − sin θ
reason
90° + θ : name changes;
angle in quadrant 2;
cos (90° + θ) < 0 . . . CAST
[ Q 1.14 ]
1.15 tan (270° − θ) = cot θ
reason
270° − θ : name changes;
angle in quadrant 3;
tan (270° − θ) > 0 . . . CAST
[ Q 1.15 ]
1.16 cot (270° + θ) = − tan θ
reason
270° + θ : name changes;
angle in quadrant 4;
cot (270° + θ) < 0 . . . CAST
[ Q 1.16 ]
1.17 cosec (90° − θ) = sec θ
reason
90° − θ : name changes;
angle in quadrant 1;
cosec (90° − θ) > 0 . . . CAST
[ Q 1.17 ]
1.18 sec (90° + θ) = − cosec θ
reason
90° + θ : name changes;
angle in quadrant 2;
sec (90° + θ) < 0 . . . CAST
[ Q 1.18 ]
1.19 tan (90° + θ) = − cot θnbsp;
[ Q 1.19 ]
1.20 sec (270° + θ) = cosec θ
[ Q 1.20 ]
1.21 sin (270° − θ) = − cos θ
[ Q 1.21 ]
1.22 cos (270° + θ) = − sin θ
[ Q 1.22 ]
1.23 sin (90° + θ) = cos θ
[ Q 1.23 ]
1.24 cos (90° − θ) = sin θ
[ Q 1.24 ]
$$ \hspace*{2 mm}\mathrm{2.1\kern3mm\ sin\ 70°\ = sin\ (90° − 20°)\kern2mm\ } $$
$$ \hspace*{24 mm}\mathrm{= cos\ 20°\kern2mm\ } $$
[ Q 2.1 ]
$$ \hspace*{2 mm}\mathrm{2.2\kern3mm\ cos\ 110°\ = cos\ (90° + 20°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= − sin\ 20°\kern2mm\ } $$
[ Q 2.2 ]
$$ \hspace*{2 mm}\mathrm{2.3\kern3mm\ cosec\ 200°\ = cosec\ (180° + 20°)\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{= − cosec 20°\kern2mm\ } $$
[ Q 2.3 ]
$$ \hspace*{2 mm}\mathrm{2.4\kern3mm\ sec\ 340°\ = sec\ (360° − 20°)\kern2mm\ } $$
$$ \hspace*{27 mm}\mathrm{= sec 20°\kern2mm\ } $$
[ Q 2.4 ]
$$ \hspace*{2 mm}\mathrm{2.5\kern3mm\ cot\ 250°\ = cot\ (270° − 20°)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{= tan 20°\kern2mm\ } $$
[ Q 2.5 ]
$$ \hspace*{2 mm}\mathrm{2.6\kern3mm\ tan\ 380°\ = tan\ (360° + 20°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= tan 20°\kern2mm\ } $$
[ Q 2.6 ]
$$ \hspace*{2 mm}\mathrm{2.7\kern3mm\ sin\ 200°\ = sin\ (180° + 20°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= − sin 20°\kern2mm\ } $$
[ Q 2.7 ]
$$ \hspace*{2 mm}\mathrm{2.8\kern3mm\ cos\ 160°\ = cos\ (180° − 20°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= − cos 20°\kern2mm\ } $$
[ Q 2.8 ]
$$ \hspace*{2 mm}\mathrm{2.9\kern3mm\ tan\ 290°\ = tan\ (270° + 20°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= − cot 20°\kern2mm\ } $$
[ Q 2.9 ]
$$ \hspace*{2 mm}\mathrm{2.10\kern3mm\ cosec\ 70°\ = cosec\ (90° − 20°)\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{= sec 20°\kern2mm\ } $$
[ Q 2.10 ]
$$ \hspace*{2 mm}\mathrm{2.11\kern3mm\ cot\ 110°\ = cot\ (90° + 20°)\kern2mm\ } $$
$$ \hspace*{27 mm}\mathrm{= − tan 20°\kern2mm\ } $$
[ Q 2.11 ]
$$ \hspace*{2 mm}\mathrm{2.12\kern3mm\ sec\ 250°\ = sec\ (270° − 20°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= − cosec 20°\kern2mm\ } $$
[ Q 2.12 ]
$$ \hspace*{2 mm}\mathrm{2.13\kern3mm\ cot\ 200°\ = cot\ (180° + 20°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= − cot 20°\kern2mm\ } $$
[ Q 2.13 ]
$$ \hspace*{2 mm}\mathrm{2.14\kern3mm\ cosec\ 250°\ = cosec\ (270° − 20°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= − sec 20°\kern2mm\ } $$
[ Q 2.14 ]
$$ \hspace*{2 mm}\mathrm{2.15\kern3mm\ sin\ 110°\ = sin\ (90° + 20°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= cos 20°\kern2mm\ } $$
[ Q 2.15 ]
$$ \hspace*{2 mm}\mathrm{2.16\kern3mm\ cos\ 70°\ = cos\ (90° − 20°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= sin 20°\kern2mm\ } $$
[ Q 2.16 ]
$$ \hspace*{2 mm}\mathrm{2.17\kern3mm\ tan\ 110°\ = tan\ (90° + 20°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= − cot 20°\kern2mm\ } $$
[ Q 2.17 ]
$$ \hspace*{2 mm}\mathrm{2.18\kern3mm\ sec\ 70°\ = sec\ (90° − 20°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= cosec 20°\kern2mm\ } $$
[ Q 2.18 ]
$$ \hspace*{2 mm}\mathrm{3.1\kern3mm\ cos\ 50°\ = cos\ (90° − 40°)\kern2mm\ } $$
$$ \hspace*{24 mm}\mathrm{= sin 40°\kern2mm\ } $$
[ Q 3.1 ]
$$ \hspace*{2 mm}\mathrm{3.2\kern3mm\ cosec\ 320°\ = cosec\ (360° − 40°)\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{= −\ cosec 40°\kern2mm\ } $$
[ Q 3.2 ]
$$ \hspace*{2 mm}\mathrm{3.3\kern3mm\ cot\ 310°\ = cot\ (270° + 40°)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{= − tan 40°\kern2mm\ } $$
[ Q 3.3 ]
$$ \hspace*{2 mm}\mathrm{3.4\kern3mm\ tan\ 130°\ = tan\ (90° + 40°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= − cot 40°\kern2mm\ } $$
[ Q 3.4 ]
$$ \hspace*{2 mm}\mathrm{3.5\kern3mm\ sin\ 230°\ = sin\ (270° − 40°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= −\ cos 40°\kern2mm\ } $$
[ Q 3.5 ]
$$ \hspace*{2 mm}\mathrm{3.6\kern3mm\ sec\ 400°\ = sec\ (360° + 40°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= sec 40°\kern2mm\ } $$
[ Q 3.6 ]
$$ \hspace*{2 mm}\mathrm{3.7\kern3mm\ cos\ 310°\ = cos\ (270° + 40°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= sin 40°\kern2mm\ } $$
[ Q 3.7 ]
$$ \hspace*{2 mm}\mathrm{3.8\kern3mm\ sec\ 230°\ = sec\ (270° − 40°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= − cosec 40°\kern2mm\ } $$
[ Q 3.8 ]
$$ \hspace*{2 mm}\mathrm{3.9\kern3mm\ cosec\ 130°\ = cosec\ (90° + 40°)\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{= sec 40°\kern2mm\ } $$
[ Q 3.9 ]
$$ \hspace*{2 mm}\mathrm{3.10\kern3mm\ sin\ 220°\ = sin\ (180° + 40°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= −\ sin 40°\kern2mm\ } $$
[ Q 3.10 ]
$$ \hspace*{2 mm}\mathrm{3.11\kern3mm\ cos\ 130°\ = cos\ (90° + 40°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= −\ sin 40°\kern2mm\ } $$
[ Q 3.11 ]
$$ \hspace*{2 mm}\mathrm{3.12\kern3mm\ tan\ 230°\ = tan\ (270° − 40°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= cot\ 40°\kern2mm\ } $$
[ Q 3.12 ]
$$ \hspace*{2 mm}\mathrm{3.13\kern3mm\ cot\ 50°\ = cot\ (90° − 40°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= tan\ 40°\kern2mm\ } $$
[ Q 3.13 ]
$$ \hspace*{2 mm}\mathrm{3.14\kern3mm\ cosec\ 310°\ = cosec\ (270° + 40°)\kern2mm\ } $$
$$ \hspace*{32 mm}\mathrm{= −\ sec\ 40°\kern2mm\ } $$
[ Q 3.14 ]
$$ \hspace*{2 mm}\mathrm{3.15\kern3mm\ sec\ 140°\ = sec\ (180° − 40°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= −\ sec\ 40°\kern2mm\ } $$
[ Q 3.15 ]
$$ \hspace*{2 mm}\mathrm{3.16\kern3mm\ cos\ 230°\ = cos\ (270° − 40°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= −\ sin\ 40°\kern2mm\ } $$
[ Q 3.16 ]
$$ \hspace*{2 mm}\mathrm{3.17\kern3mm\ sin\ 320°\ = sin\ (360° − 40°)\kern2mm\ } $$
$$ \hspace*{27 mm}\mathrm{= −\ sin\ 40°\kern2mm\ } $$
[ Q 3.17 ]
$$ \hspace*{2 mm}\mathrm{3.18\kern3mm\ tan\ 140°\ = tan\ (180° − 40°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= −\ tan\ 40°\kern2mm\ } $$
[ Q 3.18 ]
$$ \hspace*{2 mm}\mathrm{4.1\kern3mm\ sin\ 155°\ = sin\ (180° − 25°)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{= sin\ 25°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{sin\ 155°\ = sin\ (90° + 65°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= cos\ 65°\kern2mm\ } $$
[ Q 4.1 ]
$$ \hspace*{2 mm}\mathrm{4.2\kern3mm\ cosec\ 222°\ = cosec\ (180° + 42°)\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{= −\ cosec\ 42°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{cosec\ 222°\ = cosec\ (270° − 48°)\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{= −\ sec\ 48°\kern2mm\ } $$
[ Q 4.2 ]
$$ \hspace*{2 mm}\mathrm{4.3\kern3mm\ tan\ 145°\ = tan\ (180° − 35°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= −\ tan\ 35°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{tan\ 145°\ = tan\ (90° + 55°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= −\ cot\ 55°\kern2mm\ } $$
[ Q 4.3 ]
$$ \hspace*{2 mm}\mathrm{4.4\kern3mm\ cot\ 213°\ = cot\ (180° + 33°)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{= cot\ 33°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{cot\ 213°\ = cot\ (270° − 57°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= tan\ 57°\kern2mm\ } $$
[ Q 4.4 ]
$$ \hspace*{2 mm}\mathrm{4.5\kern3mm\ sec\ 298°\ = sec\ (360° − 62°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= sec\ 62°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{sec\ 298°\ = sec\ (270° + 28°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= cosec\ 28°\kern2mm\ } $$
[ Q 4.5 ]
$$ \hspace*{2 mm}\mathrm{4.6\kern3mm\ cosec\ 124°\ = cosec\ (180° − 56°)\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{= cosec\ 56°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{cosec\ 124°\ = cosec\ (90° + 34°)\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{= sec\ 34°\kern2mm\ } $$
[ Q 4.6 ]
$$ \hspace*{2 mm}\mathrm{4.7\kern3mm\ sin\ 138°\ = sin\ (180° − 42°)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{= sin\ 42°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{sin\ 138°\ = sin\ (90° + 48°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= cos\ 48°\kern2mm\ } $$
[ Q 4.7 ]
$$ \hspace*{2 mm}\mathrm{4.8\kern3mm\ sec\ 318°\ = sec\ (360° − 42°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= sec\ 42°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{sec\ 318°\ = sec\ (270° + 48°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= cosec\ 48°\kern2mm\ } $$
[ Q 4.8 ]
$$ \hspace*{2 mm}\mathrm{4.9\kern3mm\ cot\ 118°\ = cot\ (180° *minus; 62°)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{= −\ cot\ 62°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{cot\ 118°\ = cot\ (90° + 28°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= −\ tan\ 28°\kern2mm\ } $$
[ Q 4.9 ]
$$ \hspace*{2 mm}\mathrm{4.10\kern3mm\ cosec\ 346°\ = cosec\ (360° − 14°)\kern2mm\ } $$
$$ \hspace*{32 mm}\mathrm{= −\ cosec\ 14°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{cosec\ 346°\ = cosec\ (270° + 76°)\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{= −\ sec\ 76°\kern2mm\ } $$
[ Q 4.10 ]
$$ \hspace*{2 mm}\mathrm{4.11\kern3mm\ cot\ 288°\ = cot\ (360° − 72°)\kern2mm\ } $$
$$ \hspace*{27 mm}\mathrm{= −\ cot\ 72°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{cot\ 288°\ = cot\ (270° + 18°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= −\ tan\ 18°\kern2mm\ } $$
[ Q 4.11 ]
$$ \hspace*{2 mm}\mathrm{4.12\kern3mm\ tan\ 113°\ = tan\ (180° − 67°)\kern2mm\ } $$
$$ \hspace*{27 mm}\mathrm{= −\ tan\ 67°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{tan\ 113°\ = tan\ (90° + 23°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= −\ cot\ 23°\kern2mm\ } $$
[ Q 4.12 ]
$$ \hspace*{2 mm}\mathrm{4.13\kern3mm\ cos\ 308°\ = cos\ (360° − 52°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= cos\ 52°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{cos\ 308°\ = cos\ (270° + 88°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= sin\ 38°\kern2mm\ } $$
[ Q 4.13 ]
$$ \hspace*{2 mm}\mathrm{4.14\kern3mm\ sin\ 150°\ = sin\ (180° − 30°)\kern2mm\ } $$
$$ \hspace*{27 mm}\mathrm{= sin\ 30°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{sin\ 150°\ = sin\ (90° + 60°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= cos\ 60°\kern2mm\ } $$
[ Q 4.14 ]
$$ \hspace*{2 mm}\mathrm{4.15\kern3mm\ sec\ 218°\ = sec\ (180° + 38°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= −\ sec\ 38°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{sec\ 218°\ = sec\ (270° − 52°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= −\ cosec\ 52°\kern2mm\ } $$
[ Q 4.15 ]
$$ \hspace*{2 mm}\mathrm{4.16\kern3mm\ cot\ 128°\ = cot\ (180° − 52°)\kern2mm\ } $$
$$ \hspace*{27 mm}\mathrm{= −\ cot\ 52°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{cot\ 128°\ = cot\ (90° + 38°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= −\ tan\ 38°\kern2mm\ } $$
[ Q 4.16 ]
$$ \hspace*{2 mm}\mathrm{4.17\kern3mm\ cosec\ 232°\ = cosec\ (180° + 52°)\kern2mm\ } $$
$$ \hspace*{32 mm}\mathrm{= −\ cosec\ 52°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{cosec\ 232°\ = cosec\ (180° + 52°)\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{= −\ sec\ 52°\kern2mm\ } $$
[ Q 4.17 ]
$$ \hspace*{2 mm}\mathrm{4.18\kern3mm\ cos\ 142°\ = cos\ (180° − 38°)\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= −\ cos\ 38°\kern2mm\ } $$
$$ \hspace*{37 mm}\mathrm{\bold{OR}\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{cos\ 142°\ = cos\ (90° + 52°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= −\ sin\ 52°\kern2mm\ } $$
[ Q 4.18 ]
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°
$$ \hspace*{2 mm}\mathrm{5.1\kern3mm\ cos\ 67°\ = cos\ (90° − 23°)\kern2mm\ } $$
$$ \hspace*{24 mm}\mathrm{= sin\ 23°\kern2mm\ } $$
$$ \hspace*{24 mm}\mathrm{= p\kern2mm\ } $$
[ Q 5.1 ]
$$ \hspace*{2 mm}\mathrm{5.2\kern3mm\ sin\ 157°\ = sin\ (180° − 23°)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{= sin\ 23°\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{= p\kern2mm\ } $$
[ Q 5.2 ]
$$ \hspace*{2 mm}\mathrm{5.3\kern3mm\ cos\ 293°\ = cos\ (270° + 23°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= sin\ 23°\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= p\kern2mm\ } $$
[ Q 5.3 ]
$$ \hspace*{2 mm}\mathrm{5.4\kern3mm\ sin\ 203°\ = sin\ (180° + 23°)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{= −\ sin\ 23°\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{= −\ p\kern2mm\ } $$
[ Q 5.4 ]
$$ \hspace*{2 mm}\mathrm{5.5\kern3mm\ cos\ 113°\ = cos\ (90° + 23°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= −\ sin\ 23°\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= −\ p\kern2mm\ } $$
[ Q 5.5 ]
$$ \hspace*{2 mm}\mathrm{5.6\kern3mm\ sin\ 337°\ = sin\ (360° − 23°)\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{= −\ sin\ 23°\kern2mm\ } $$
$$ \hspace*{25 mm}\mathrm{= −\ p\kern2mm\ } $$
[ Q 5.6 ]
$$ \hspace*{2 mm}\mathrm{5.7\kern3mm\ sin^2\ 23°\ +\ cos^2\ 23°\ = 1\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{cos\ 23°\ = \sqrt(1 − sin^2 23°)\kern2mm\ } $$
$$ \hspace*{44 mm}\mathrm{= \sqrt(1 − p^2)\kern2mm\ } $$
[ Q 5.7 ]
$$ \hspace*{2 mm}\mathrm{5.8\kern3mm\ cos\ 247°\ = cos\ (270° − 23°)\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= −\ sin\ 23°\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= −\ p\kern2mm\ } $$
[ Q 5.8 ]
$$ \hspace*{2 mm}\mathrm{5.9\kern3mm\ cosec\ 23°\ =\ \frac{1}{sin\ 23°}\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= \frac{1}{p}\kern2mm\ } $$
[ Q 5.9 ]
$$ \hspace*{2 mm}\mathrm{5.10\kern3mm\ tan\ 23°\ = \frac{sin\ 23°}{cos\ 23°}\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= \frac{p}{\sqrt(1 − p^2)}\kern2mm\ } $$
$$ \hspace*{26 mm}\mathrm{= \frac{p\ \ \sqrt(1 − p^2)}{1 − p^2}\kern2mm\ } $$
[ Q 5.10 ]
$$ \hspace*{2 mm}\mathrm{6.1\kern3mm\ sec\ (180° + x)\ .\ cosec\ (180° − x)\ .\ cos\ (360° − x)\ .\ sin\ (180° + x)\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= (−\ sec\ x)\ .\ (cosec\ x)\ .\ (cos\ x)\ .\ (−\ sin\ x)\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{1}{cos\ x}\ \times\ \frac{1}{sin\ x}\ \times\ cos\ x\ \times\ sin\ x\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= 1\kern2mm\ } $$
[ Q 6.1 ]
$$ \hspace*{2 mm}\mathrm{6.2\kern3mm\ tan\ (180° + x)\ .\ cos\ (360° − x)\ .\ sec\ (90° + x)\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= (tan\ x)\ .\ (cos\ x)\ .\ (−\ cosec\ x)\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −\ \frac{sin\ x}{cos\ x}\ \times\ cos\ x\ \times \frac{1}{sin\ x}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −\ 1\kern2mm\ } $$
[ Q 6.2 ]
$$ \hspace*{2 mm}\mathrm{6.3\kern3mm\ tan\ (180° + x)\ −\ tan\ (180° − x)\ + tan\ (360° + x)\ −\ tan\ (360° − x)\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= (tan\ x)\ −\ (−\ tan\ x)\ + (tan\ x)\ −\ (−\ tan\ x)\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −\ 4 tan\ x\kern2mm\ } $$
[ Q 6.3 ]
$$ \hspace*{2 mm}\mathrm{6.4\kern3mm\ sin\ (180° − x)\ .\ cos\ (90° − x)\ −\ cos\ (180° + x)\ .\ cos\ (360° − x)\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= (sin\ x)\ .\ (sin\ x)\ −\ (−\ cos\ x)\ .\ (cos\ x)\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= sin^2\ x)\ + cos^2\ x\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= 1\kern2mm\ } $$
[ Q 6.4 ]
$$ \hspace*{2 mm}\mathrm{6.5\kern3mm\ sin\ (90° + x)\ .\ cos\ (90° − x)\ + cos\ (90° + x)\ .\ sin\ (90° − x)\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= (cos\ x)\ .\ (sin\ x)\ + (−\ sin\ x)\ .\ (cos\ x)\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= 0\kern2mm\ } $$
[ Q 6.5 ]
$$ \hspace*{2 mm}\mathrm{6.6\kern3mm\frac{sin\ (360° − x)\ .\ tan\ (180° + x)}{cos\ x\ .\ cot\ (90° − x)\ .\ sec\ (180° + x)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{(− sin\ x)\ .\ (tan\ x)}{cos\ x\ .\ (tan\ x)\ .\ \Big(−\ \frac{1}{cos\ x}\Big)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= sin\ x\kern2mm\ } $$
[ Q 6.6 ]
$$ \hspace*{2 mm}\mathrm{6.7\kern3mm\frac{cos\ (180° + θ)\ .\ sin\ (180° − θ)}{cos\ (360° − θ)\ .\ tan\ (360° − θ)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{(− cos\ θ)\ .\ (sin\ θ)}{cos\ θ\ .\ (−\ tan\ θ)\ .\ \Big(−\ \frac{1}{cos\ θ}\Big)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{sin\ θ}{1} \times \frac{cos\ θ}{sin\ θ}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= cos\ θ\kern2mm\ } $$
[ Q 6.7 ]
$$ \hspace*{2 mm}\mathrm{6.8\kern3mm\frac{sin\ (360° + x)\ \div\ cos\ (90deg; − x)\ \div\ sin\ (180° − x)}{tan\ (180° + x)\ \div\ tan\ (360° + x)\ \div\ cos\ (270° − x)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{(sin\ x)\ \div (sin\ x)\ \div\ (sin\ x)}{(tan\ x)\ \div\ (tan\ x)\ \div\ (−\ sin\ x)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −\ \frac{1\ \div (sin\ x)}{1 \div\ (sin\ x)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −\ 1\kern2mm\ } $$
[ Q 6.8 ]
$$ \hspace*{2 mm}\mathrm{6.9\kern3mm\frac{cos^2\ (180° − x)}{sin\ (360° − x)\ .\ sin\ (90° − x)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{(− cos\ x)^2}{(−\ sin\ x)\ .\ (cos\ x)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −\ \frac{cos\ x}{sin\ x}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −\ cot\ x\kern2mm\ } $$
[ Q 6.9 ]
$$ \hspace*{2 mm}\mathrm{6.10\kern3mm\frac{cos\ (90° + α)\ .\ sin\ (−\ α)}{sin\ (180° + x)\ .\ tan\ (360° − α)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{(−\ sin\ α)\ .\ (−\ sin\ α)}{(−\ sin\ α)\ .\ (− tan\ α)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{sin\ α}{1} \times \frac{cos\ α}{sin\ α}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= cos\ α\kern2mm\ } $$
[ Q 6.10 ]
$$ \hspace*{2 mm}\mathrm{6.11\kern3mm\frac{cos\ (−\ β)\ .\ sin\ (180° + β)}{tan\ (β − 180°)\ .\ sin\ (β + 90°)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{(cos\ α)\ .\ (−\ sin\ α)}{(−\ tan\ (180° − β))\ .\ (cos\ β)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{cos\ β\ .\ sin\ β}{(−\ tan\ β)\ .\ cos\ β}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −\ \frac{sin\ β}{1}\ \times \frac{cos\ β}{sin\ β}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −\ cos\ β\kern2mm\ } $$
[ Q 6.11 ]
$$ \hspace*{2 mm}\mathrm{6.12\kern3mm\frac{sin\ (−\ α)\ .\ sec\ (−\ α)\ .\ tan\ (180° − α)}{sec\ (360° − α)\ .\ cos\ (270^deg; − α)\ .\ tan\ (180° + α)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{(−\ sin\ α)\ .\ (sec\ α)\ .\ (−\ tan\ α)}{(sec\ α)\ .\ (− sin\ α)\ .\ (tan\ α)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −\ 1\kern2mm\ } $$
[ Q 6.12 ]
$$ \hspace*{2 mm}\mathrm{7.1\kern3mm\ sin\ 320°\ .\ tan\ 140°\ .\ cot\ 220°\ .\ sec\ 130°\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= sin\ (360° − 40°) .\ tan\ (180° − 40°)\ .\ cot\ (180° + 40°)\ .\ sec\ (90° + 40°)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= (−\ sin\ 40°) .\ (−\ tan\ 40°)\ .\ (cot\ 40°)\ .\ (−\ cosec\ 40°)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= −\ sin\ 40° \times \frac{sin\ 40°}{cos\ 40°} \times \frac{cos\ 40°}{sin\ 40°} \times \frac{1}{sin\ 40°}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= −\ 1\kern2mm\ } $$
[ Q 7.1 ]
$$ \hspace*{2 mm}\mathrm{7.2\kern3mm\ sin\ 200°\ .\ sec\ 110°\ + cos\ 160°\ .\ cosec\ 110°\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= sin\ (180° − 20°) .\ sec\ (90° + 20°)\ + cos\ (180° − 20°)\ .\ cosec\ (90° + 20°)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= (−\ sin\ 20°) .\ (−\ cosec\ 20°)\ + (−\ cos\ 20°)\ .\ (sec\ 20°)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= 1 + (−\ 1)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= 0\kern2mm\ } $$
[ Q 7.2 ]
$$ \hspace*{2 mm}\mathrm{7.3\kern3mm\ cos\ 290°\ .\ tan\ 200°\ −\ cos\ 160°\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= cos\ (270° + 20°)\ .\ tan\ (180° + 20°)\ −\ cos\ (180° − 20°)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= (−\ sin\ 20°)\ .\ (−\ tan\ 20°)\ −\ (−\ cos\ 20°)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= sin\ 20°\ \times \frac{sin 20°}{cos\ 20°}\ + cos\ 20°\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= \frac{sin^2\ 20°}{cos\ 20°}\ \ +\ \ \frac{cos^2\ 20°}{cos\ 20°}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= \frac{1}{cos\ 20°}\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= sec\ 20°\kern2mm\ } $$
[ Q 7.3 ]
$$ \hspace*{2 mm}\mathrm{7.4\kern3mm\ cos\ 63°\ .\ cosec\ 207°\ .\ cot\ 27°\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{= cos\ (90° − 27°)\ .\ cosec\ (180° + 27°)\ .\ cot\ 27°\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{= (sin\ 27°)\ .\ (−\ cosec\ 27°)\ .\ (cot\ 27°)\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{= −\ cot\ 27°\kern2mm\ } $$
[ Q 7.4 ]
$$ \hspace*{2 mm}\mathrm{7.5\kern3mm\ sin\ 125°\ .\ cot\ 145°\ .\ cosec\ 55°\ .\ sin\ 215°\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= sin\ (90° + 35°)\ .\ cot\ (180°: − 35°)\ .\ cosec\ (90° − 35°)\ .\ sin\ (180° + 35°)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= cos\ 35°\ .\ (−\ cot\ 35°)\ .\ sec\ 35°\ .\ (−\ sin\ 35°)\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= cos\ 35°\ \times \frac{cos\ 35°}{sin\ 35°}\ \times \frac{1}{cos\ 35°}\ \times sin\ 35°\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{= cos\ 35°\kern2mm\ } $$
[ Q 7.5 ]
$$ \hspace*{2 mm}\mathrm{7.6\kern3mm\frac{cosec\ 110°}{2\ sec\ 20°}\kern2mm\ =\frac{cosec\ (90° + 20°)}{2\ sec\ 20°}\kern2mm\ } $$
$$ \hspace*{31 mm}\mathrm{= \frac{sec\ (90° + 20°)}{2\ sec\ 20°}\kern2mm\ } $$
$$ \hspace*{31 mm}\mathrm{= \frac{1}{2}\kern2mm\ } $$
[ Q 7.6 ]
$$ \hspace*{2 mm}\mathrm{7.7\kern3mm\frac{tan\ 340°\ .\ sin\ 72°}{cos\ 18°\ .\ cot\ 70°}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{tan\ (270° + 70°)\ .\ sin\ 72°}{cos\ (90° − 72°)18°\ .\ cot\ 70°}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{tan\ (270° + 70°)\ .\ sin\ 72°}{cos\ (90° − 72°)18°\ .\ cot\ 70°}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{(−\ cot\ 70°)\ .\ sin\ 72°}{sin\ 72°)18°\ .\ cot\ 70°}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −\ 1\kern2mm\ } $$
[ Q 7.7 ]
$$ \hspace*{2 mm}\mathrm{7.8\kern3mm\frac{sec\ 185°\ .\ sin\ 78°}{cos\ 348°\ .\ sec\ 175°}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{sec\ (180° + 5°)\ .\ sin\ 78°}{cos\ (270° + 78°)\ .\ sec\ (170° + 5°}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{(−\ sec\ 5°)\ .\ sin\ 78°}{(sn\ 78°)\ .\ (−\ sec\ 5°)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= 1\kern2mm\ } $$
[ Q 7.8 ]
$$ \hspace*{2 mm}\mathrm{7.9\kern3mm\frac{3\ tan\ 65°\ .\ cosec\ 160°}{4\ cosec\ 200°\ .\ cot\ 25°}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{3\ tan\ 65°\ .\ cosec\ (180° − 20°)}{4\ cosec\ (180°' + 20°)\ .\ cot\ (90° − 65°)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= \frac{3\ tan\ 65°\ .\ (cosec\ 20°)}{4\ (−\ cosec\ 20°)\ .\ (tan\ 65°)}\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{= −\ \frac{3}{4}\kern2mm\ } $$
[ Q 7.9 ]
$$ \hspace*{2 mm}\mathrm{7.10\kern3mm\frac{cot\ 27°\ .\ tan\ 153°}{sin\ 27°\ .\ cos\ 117°\ −\ cosec\ 207°\ .\ sin\ 333°}\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{= \frac{cot\ 27°\ .\ tan\ (180° − 27°}{sin\ 27°\ .\ cos\ (90° + 27°)\ −\ cosec\ (180°' + 27°)\ .\ sin\ (360° − 27°)}\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{= \frac{cot\ 27°\ .\ (−\ tan\ 27°)}{sin\ 27°\ .\ (−\ sin\ 27°)\ −\ (−\ cosec\ 27°)\ .\ (−\ sin\ 27°)}\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{= \frac{−\ 1}{−\ sin^2\ 27° −\ 1}\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{= sin^2\ 27° + 1\kern2mm\ } $$
[ Q 7.10 ]
$$ \hspace*{2 mm}\mathrm{7.11\kern3mm\frac{1}{cot\ 52°}\ \times \ cosec\ 128°\ + sec\ 232°\ .\ cos^2\ 142°\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{= tan\ 52°\ \times \ cosec\ (180° − 52°)\ + sec\ (180° + 52°)\ .\ (cos\ (90° + 52°))^2\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{= tan\ 52°\ \times \ cosec\ 52°\ + (−\ sec\ 52°)\ .\ (−\ sin\ 52°)\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{= \frac{sin\ 52°}{cos\ 52°}\ \times \frac{−\ 1}{sin\ 52°}\ + \frac{−\ 1}{cos\ 52°}\ \times \frac{sin^2\ 52°}{1}\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{= \frac{1}{cos\ 52°}\ −\ \frac{sin^2\ 52°}{cos\ 52°}\ \times \frac{sin^2\ 52°}{1}\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{= \frac{1 −\ sin^2\ 52°}{cos\ 52°}\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{= \frac{cos^2\ 52°}{cos\ 52°}\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{= cos\ 52°\kern2mm\ } $$
[ Q 7.11 ]
$$ \hspace*{2 mm}\mathrm{7.12\kern3mm\frac{sin\ 108°\ .\ tan\ 72°\ .\ sin\ 162°}{tan\ 252°\ .\ cos\ (−\ 72°)\ .\ sin\ (−\ 72°)}\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{= \frac{sin\ (180° − 72°)\ .\ tan\ 72°\ .\ sin\ (90° + 72°)}{tan\ (180° + 72°)\ .\ (cos\ 72°)\ .\ (−\ sin\ 72°)}\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{= −\ \frac{sin\ 72°\ .\ tan\ 72°\ .\ cos\ 72°}{tan\ 72°\ .\ cos\ 72°\ .\ sin\ 72°}\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{= −\ 1\kern2mm\ } $$
[ Q 7.12 ]