124 GEOMETRY AND TRIGONOMETRY
- sin−^10. 2341
[
13. 54 ◦,13◦ 32 ′,
0 .236 rad]- cos−^10. 8271
[
34. 20 ◦,34◦ 12 ′,
0 .597 rad]- tan−^10. 8106
[
39. 03 ◦,39◦ 2 ′,
0 .681 rad]- sec−^11. 6214
[
51. 92 ◦,51◦ 55 ′,
0 .906 rad]- cosec−^12. 4891
[
23. 69 ◦,23◦ 41 ′,
0 .413 rad]- cot−^11. 9614
[
27. 01 ◦,27◦ 1 ′,
0 .471 rad]In Problems 15 to 18, evaluate correct to 4
significant figures.- 4 cos 56◦ 19 ′−3 sin 21◦ 57 ′ [1.097]
16.11 .5 tan 49◦ 11 ′−sin 90◦
3 cos 45◦[5.805]17.5 sin 86◦ 3 ′
3 tan 14◦ 29 ′−2 cos 31◦ 9 ′[−5.325]18.6 .4 cosec 29◦ 5 ′−sec 81◦
2 cot 12◦[0.7199]- Determine the acute angle, in degrees and
minutes, correct to the nearest minute, given
by sin−^1(
4 .32 sin 42◦ 16 ′
7. 86)[21◦ 42 ′]- If tanx= 1 .5276, determine secx, cosecx,
and cotx. (Assumexis an acute angle)
[1.8258, 1.1952, 0.6546]
In Problems 21 to 23 evaluate correct to 4
significant figures
21.(sin 34◦ 27 ′)(cos 69◦ 2 ′)
(2 tan 53◦ 39 ′)[0.07448]- 3 cot 14◦ 15 ′sec 23◦ 9 ′ [12.85]
23.cosec 27◦ 19 ′+sec 45◦ 29 ′
1 −cosec 27◦ 19 ′sec 45◦ 29 ′[−1.710]- Evaluate correct to 4 decimal places:
(a) sine (− 125 ◦) (b) tan (− 241 ◦)
(c) cos (− 49 ◦ 15 ′)
[
(a)− 0. 8192 (b)− 1. 8040
(c) 0. 6528]- Evaluate correct to 5 significant figures:
(a) cosec (− 143 ◦) (b) cot (− 252 ◦)
(c) sec (− (^67) [◦ 22 ′)
(a)− 1 .6616 (b)− 0. 32492
(c) 2. 5985
]
12.7 Sine and cosine rules
To‘solve a triangle’means ‘to find the values of
unknown sides and angles’. If a triangle isright
angled, trigonometric ratios and the theorem of
Pythagoras may be used for its solution, as shown
in Section 12.4. However, for anon-right-angled
triangle, trigonometric ratios and Pythagoras’ the-
oremcannotbe used. Instead, two rules, called the
sine ruleand thecosine rule, are used.
Sine rule
With reference to triangleABCof Fig. 12.19, the
sine rulestates:
a
sinA
b
sinB
c
sinC
Figure 12.19
The rule may be used only when:
(i) 1 side and any 2 angles are initially given, or
(ii) 2 sides and an angle (not the included angle) are
initially given.
Cosine rule
With reference to triangleABCof Fig. 12.19, the
cosine rulestates: