Basic Engineering Mathematics, Fifth Edition

Introduction to trigonometry 185

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)

7

6

4

3

2



Figure 21.13

(a) PointsAandBare shown in Figure 21.13(a).

In Figure 21.13(b), the horizontal and vertical

linesAC andBCare constructed. SinceABC is

a right-angled triangle, andAC=( 8 − 2 )=6and

BC=( 7 − 3 )=4, by Pythagoras’ theorem,

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

and AB=

√

62 + 42 =

√

52

=7.211correct to 3

decimal places.

(b) The gradient of AB is given by tanθ,i.e.

gradient=tanθ=

BC

AC

=

4

6

=

2

3

Now try the following Practice Exercise

PracticeExercise 83 Trigonometric ratios

(answers on page 349)

1. Sketch a triangleXY Zsuch that
   ∠Y= 90 ◦, XY=9cm and YZ=40cm.
   Determine sinZ,cosZ,tanXand cosX.


2. In triangleABCshown in Figure 21.14, find
   sinA,cosA,tanA,sinB,cosBand tanB.

B

C

(^53)
A
Figure 21.14

3. If cosA=
   15
   17

,findsinAand tanA, in fraction

form.

4. If tanX=

15

112

,findsinXand cosX, in frac-

tion form.

5. For the right-angled triangle shown in
   Figure 21.15, find (a) sinα(b) cosθ(c) tanθ.



 17

8

15

Figure 21.15

6. If tanθ=

7

24

,findsinθand cosθin fraction

form.

7. PointPlies at co-ordinate(− 3 , 1 )and point
   Qat( 5 ,− 4 ). Determine
   (a) the distancePQ.
   (b) the gradient of the straight linePQ.

21.4 Evaluating trigonometric ratios

of acute angles

The easiest way to evaluate trigonometric ratios of any

angle is to use a calculator. Use a calculator to check the

following (each correct to 4 decimal places).

sin29◦=0.4848 sin53. 62 ◦=0.8051

cos67◦=0.3907 cos83. 57 ◦=0.1120

tan34◦=0.6745 tan67. 83 ◦=2.4541

sin67◦ 43 ′=sin67

43

60

◦

=sin67. 7166666 ...◦=0.9253

cos13◦ 28 ′=cos13

28

60

◦

=cos13. 466666 ...◦=0.9725

tan56◦ 54 ′=tan56

54

60

◦

=tan56. 90 ◦=1.5340

If we know the value ofa trigonometricratio and need to

find the angle we use theinverse functionon our calcu-

lators. For example, using shift and sin on our calculator

gives sin−^1 (

If, for example, we know the sine of an angle is 0.5 then

the value of the angle is given by

sin−^10. 5 = 30 ◦(Check that sin30◦= 0. 5 )

Similarly, if

cosθ= 0 .4371 thenθ=cos−^10. 4371 =64.08◦