Higher Engineering Mathematics, Sixth Edition

100 Higher Engineering Mathematics

B

A

0 2 4

(a)

(b)

68

8

f(x)

7 6 4 3 2 B

A C

0 2 468

8

f(x)

6

4

2

Figure 11.8

Now try the following exercise

Exercise 45 Further problems on

trigonometric ratios of acute angles

1. In triangleABC shown in Fig. 11.9, find
   sinA,cosA,tanA,sinB,cosBand tanB.
   ⎡
   ⎣

sinA=^35 ,cosA=^45 ,tanA=^34

sinB=^45 ,cosB=^35 ,tanB=^43

⎤

⎦

B

C

(^53)
A
Figure 11.9

2. If cosA=

15

17

find sinAand tanA, in fraction

form.

[

sinA=

8

17

,tanA=

8

15

]

3. For the right-angled triangle shown in
   Fig. 11.10, find:

(a) sinα (b) cosθ (c) tanθ

[

(a)

15

17

(b)

15

17

(c)

8

15

]



 17

8

15

Figure 11.10

4. PointPlies at co-ordinate (−3, 1) and point
   Qat (5,−4). Determine
   (a) the distancePQ
   (b) the gradient of the straight linePQand
   (c) the anglePQmakes with the horizontal.
   [(a) 9.434 (b)− 0 .625 (c) 32◦]

11.4 Evaluating trigonometric ratios

The easiest method of evaluating trigonometric func-

tions of any angle is by using acalculator.

The following values, correct to 4 decimal places,

may be checked:

sine 18◦= 0 .3090, cosine 56◦= 0. 5592

sine 172◦= 0. 1392 cosine 115◦=− 0 .4226,

sine 241. 63 ◦=− 0 .8799, cosine 331. 78 ◦= 0. 8811

tangent 29◦= 0 .5543,

tangent 178◦=− 0. 0349

tangent 296. 42 ◦=− 2. 0127

To evaluate, say, sine 42◦ 23 ′ using a calculator

means finding sine42

23 ◦

60

since there are 60 minutes

in1degree.

23

60

= 0. 383 3 thus 42 ̇ ◦ 23 ′= 42. 383 ̇◦

Thus sine 42◦ 23 ′=sine 42. 383 ̇◦= 0 .6741, correct to 4

decimal places.

Similarly, cosine72◦ 38 ′=cosine72

38 ◦

60

=0.2985,

correct to 4 decimal places.

Most calculators contain only sine, cosine and tan-

gent functions. Thus to evaluate secants, cosecants and

cotangents, reciprocals need to be used. The following