Higher Engineering Mathematics

INTRODUCTION TO TRIGONOMETRY 117

B

Figure 12.5

Thus

sinX=

40

41

,tanX=

40

9

= 4

4

9

,

cosecX=

41

40

= 1

1

40

,

secX=

41

9

= 4

5

9

andcotX=

9

40

Problem 4. If sinθ= 0 .625 and cosθ= 0. 500

determine, without using trigonometric tables

or calculators, the values of cosecθ, secθ, tanθ

and cotθ.

cosecθ=

1

sinθ

=

1

0. 625

=1.60

secθ=

1

cosθ

=

1

0. 500

=2.00

tanθ=

sinθ

cosθ

=

0. 625

0. 500

=1.25

cotθ=

cosθ

sinθ

=

0. 500

0. 625

=0.80

Problem 5. PointAlies at co-ordinate (2, 3)

and pointB at (8, 7). Determine (a) the dis-

tanceAB, (b) the gradient of the straight lineAB,

and (c) the angleABmakes with the horizontal.

(a) PointsAandBare shown in Fig. 12.6(a).

In Fig. 12.6(b), the horizontal and vertical lines

ACandBCare constructed.

Since ABC is a right-angled triangle, and

AC=(8−2)=6 andBC=(7−3)=4, then by

Pythagoras’ theorem

AB^2 =AC^2 +BC^2 = 62 + 42

and AB=

√

(6^2 + 42 )=

√

52 =7.211,

correct to 3 decimal places

Figure 12.6

(b) The gradient ofABis given by tanA,

i.e.gradient=tanA=

BC

AC

=

4

6

=

2

3

(c)The angleABmakes with the horizontalis

given by tan−^123 =33.69◦.

Now try the following exercise.

Exercise 53 Further problems on trigono-

metric ratios of acute

1. In triangleABCshown in Fig. 12.7, find
   sinA, cosA, tanA, sinB, cosBand tanB.
   [
   sinA=^35 , cosA=^45 , tanA=^34

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

]

2. If cosA=

15

17

find sinAand tanA, in fraction

form.

[

sinA=

8

17

, tanA=

8

15

]