3. (i) 5√ (ii) √ (iii) √
_ 5
4. 54 cm 5. 9⋅4 km 6. 302
_ 11
_ 5
5. (i) (ii) ≠ 212 + 172
7. 9 m 8. 7, 24, 25 as 252
= 242 + 72
9. (iii) ∆ABD ≡ ∆ACD by ASA: |∠CBA| = |∠ACB|, isosceles triangle, |AB| = |AC|, isosceles triangle, |∠BAD| = |∠CAD| since |∠ADB| = |∠ADC| = 90° so third angles are equal
(iv) |BD| = 3 cm, |CD| = 3 cm (v)
10.(i)
It bisects the base (i)√
_ 97,√
_ 65
Practice questions 19.2
1. (i) Hypotenuse = |XY|; opposite = |ZY|; adjacent = |ZX|
(ii) Hypotenuse = |LN|; opposite = |LM|; adjacent = |MN|
(iii) Hypotenuse = |RS|; opposite = |ST|; adjacent = |TR|
2. (i) sin A = 8
(ii) sin B = 24 (iii) sin C = 9 tan C = 9
(v) sin E = 35 (vi) sin F = 63
3. (i)
__ 17; cos A = 15
__ 25; cos B = 7
(iv) sin D = 28 tan D = 28
__ 53; cos D = 45
__ 40
__ 53;
__ 37; cos E = 12
__ 45
(a) sin A = 4 = 4
__ 65; cos F = 16
tan B = 3
__ 3; sin B = 3
(b) sin A = 12 tan A = 12 cos B = 12
__ 13; cos A = 5
__ 4
__ 5 ; sin B = 5
(c) sin A = 8 tan A = 8 cos B = 8
__ 6; sin B = 6
__ 13;
__ 10; cos A = 6
__ 10; tan B = 6
(d) sin A = 2 tan A = 2
cos B = 2
__ 3; cos A = √
___ √
(e) sin A = 1 tan A = 1
(f) sin A = √ tan A = √ cos B = √
__ 13; tan B = 5
__ 10;
__ 8
___ 3 ;
_ 5
_ 5; sin B = √
__ 3; tan B = √
____ √
cos B = 1
__ 4; sin B = 4
____ √
____ 5 ; cos A = 2
_ 21
____ 2 ; sin B = 2
_ 21
____ 5 ; tan B = 2
_ 21
_ 17; cos A = 4
___ 2
_ 5
____ √
____ √
_ 17;
_ 17; tan B = 4
__ 5;
__ 5;
____ √
(ii) sin A = cos B in each triangle (iii) In a right-angled triangle, the sin of one of the non-right angles is equal to the cos of the other non-right angle
_ 21
4. (i) Angle R is in the top left-hand corner
(ii) cos R = a
__ c; tan R = b
__ a
_ 17;
___ 3 ;
_ 5
__ 13;
__ 12
__ 10;
__ 37; tan E = 35
__ 5; cos A = 3
__ 65; tan F = 63
__ 5; cos B = 4
__ 5; tan A
__ 5;
__ 12
__ 16
__ 17; tan A = 8
__ 41; cos C = 40
__ 41;
__ 25; tan B = 24
__ 15
__ 7
(a) √
_ 65
(b) 3√ (c) √
(a) sin X = 7 tan X = 7
_ 63
__ 4
(b) sin X = 6 tan X = 6
(c) sin X = 9 tan X = 9
6. (ii) 4⋅14 m (iii) sin R = 0.7
tan R = 0.7
7. (ii) 79⋅54 m (iii) sin S = 79.54
___ 4.14
tan S = 79.54
9. sin X = 1 10.(i)
(iii) 1 (iv) 24
(v) 24
8. sin 45°= 1 tan 45°= 1
___ √
____ 176 ; cos S = 157
_ 2; cos 45°= 1
____ 157
___ √
sin A = 8 sin B = 6
(ii) (sin A)2
__ 10
11.(i) 13 (ii) 45 units (iii) 270 units
__ 10
Practice questions 19.3
1. (i) 0⋅4067 (ii) 0⋅7071 (iii) 28⋅6363 (iv) 0⋅2126 (v) 0⋅6820 (vi) 0⋅1908 (vii) 0⋅8829 (viii) 0⋅2419 (ix) 0⋅5000 (x) 3⋅0777 (xi) 0⋅8572 (xii) 0⋅4663
2. (i) 49° (ii) 55° (iii) 8° (iv) 60° (v) 27° (vi) 32° (vii) 52° (viii) 87°
3. A = 77⋅56°; sin A = 0⋅98 4. (i) False (ii) False
5. (i)
(a) sin 10° = 0⋅1736, cos 80° = 0⋅1736
(b) sin 20° = 0⋅342, cos 70° = 0⋅342
(c) sin 30° = 0⋅5, cos 60° = 0⋅5 (ii) sin 40° = cos 50°
6. (i) 0⋅77 (ii) 0⋅17 (iii) 66°
7. (i) tan x: 0, 0⋅2, 0⋅4, 0⋅6, 0⋅8, 1⋅2, 1⋅7, 2⋅7, 5⋅7
(iii) (a) 1⋅4 (b) 45°
__ 2; cos X = √
= 64
__ 10; cos B = 8
__ 10; cos A = 6
___ 2
_ 3
__ 10;
__ 10
___ 100; (cos B)2
= 64
___ 100
_ 2;
___ 176;
__ 9; cos X = 3√
____ 3√
_ 5
__ 12; cos X = √
____ √
_ 63
____ 4.2; cos R = 4.14
____ 4.2 ;
_ 5
____ √
_ 65; cos X = 4
____ 9 ;
_ 5
____ 12 ;
_ 63
____ √
_ 65;
Practice questions 19.4
1. (i) 5⋅2 m (ii) 91⋅8 mm (iii) 19⋅5 m (iv) 14⋅5 cm (v) 9⋅7 cm (vi) 3⋅3 m
2. (i) 6⋅65 (ii) 11⋅65 (iii) 4⋅78
3. (i) 23⋅6° (ii) 35⋅1° (iii) 58°
4. w = 11⋅594; x = 12⋅433; y = 26⋅633; z = 29⋅419
5. (i) 57⋅8° (ii) 43⋅7° (iii) 56⋅1° (iv) 65⋅9° (v) 62⋅2° (vi) 72⋅9°
6. w = 49°; x = 41°; y = 34°; z = 56° 7. (i) 11 (ii) 38°
8. 16° 9. (i) 55⋅2° (ii) 9⋅9
10.(i) 2⋅76 cm (ii) 19⋅8°
11.(i) 4 cm (ii) 3⋅6 cm
12.22⋅6° 13.(i) 17
(ii) sin θ = 8 (iii) 23 (iv) 28°
__ 17 > 8
__ 15
14.(i) Third side = 21; A = 46⋅4°; other angles = 43⋅6°, 90°
(ii) Third side = 5, B = 36⋅9°; other angles = 53⋅1°, 90°
(iii) Third side = 8; C = 61⋅9°; other angles = 28⋅1°, 90°
Practice questions 19.5
1. 542 m 2. 21 m 3. 34⋅97 m 4. 37° 5. It can land safely, angle is 14° 6. 21⋅2° 7. 15⋅6° 8. (i) Clinometer (ii) 7⋅46 m
9. 319 m 10.1 744 m 11.17° 12.(i) Plane Q = 1 435 km; Plane T = 980 km
(ii) 1 738 km
13.52 m 14.397 m 15.243 m 16.(i) 314 m (ii) 2⋅48 km
17.(i) True (ii) False (iii) False (iv) True (v) False (vi) False
Answer key 387
__ 17; cos θ = 15
__ 17
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