Higher Engineering Mathematics

FUNCTIONS AND THEIR CURVES 203

C

(c) arccos

(

−

√

3

2

)

≡cos−^1

(

−

√

3

2

)

= 150 ◦

=

5 π

6

rador2.6180 rad

(d) arccosec(

√

2)=arcsin

(

1

√

2

)

≡sin−^1

(

1

√

2

)

= 45 ◦

=

π

4

rador0.7854 rad

Problem 7. Evaluate (in radians), correct to

3 decimal places: sin−^10. 30 +cos−^10. 65

sin−^10. 30 = 17. 4576 ◦= 0 .3047 rad

cos−^10. 65 = 49. 4584 ◦= 0 .8632 rad

Hence sin−^10. 30 +cos−^10. 65
= 0. 3047 + 0. 8632 =1.168, correct to 3 decimal
places.

Now try the following exercise.

Exercise 87 Further problems on inverse

functions

Determine the inverse of the functions given in

Problems 1 to 4.

1.f(x)=x+1[f−^1 (x)=x−1]

2.f(x)= 5 x− 1

[

f−^1 (x)=^15 (x+1)

]

3.f(x)=x^3 +1[f−^1 (x)=^3

√

x−1]

4.f(x)=

1

x

+ 2

[

f−^1 (x)=

1

x− 2

]

Determine the principal value of the inverse

functions in Problems 5 to 11.

5. sin−^1 (−1)

[

−

π

2

or− 1 .5708 rad

]

6. cos−^10. 5

[π

3

or 1.0472 rad

]

7. tan−^11

[π

4

or 0.7854 rad

]

8. cot−^1 2[ 0 .4636 rad]


9. cosec−^12 .5[ 0 .4115 rad]
   10. sec−^11 .5[ 0 .8411 rad]
   11. sin−^1

(

1

√

2

) [

π

4

or 0.7854 rad

]

12. Evaluatex, correct to 3 decimal places:

x=sin−^1

1

3

+cos−^1

4

5

−tan−^1

8

9

[0.257]

13. Evaluatey, correct to 4 significant figures:

y=3 sec−^1

√

2 −4 cosec−^1

√

2

+5 cot−^12 [1.533]

19.7 Asymptotes

If a table of values for the functiony=

x+ 2

x+ 1

is

drawn up for various values ofxand thenyplotted

againstx, the graph would be as shown in Fig. 19.32.

The straight linesAB, i.e.x=−1, andCD, i.e.y=1,

are known asasymptotes.

An asymptote to a curve is defined as a straight

line to which the curve approaches as the distance

from the origin increases. Alternatively, an asymp-

tote can be considered as a tangent to the curve at

infinity.

Asymptotes parallel to thex- andy-axes

There is a simple rule which enables asymptotes par-

allel to thex- andy-axis to be determined. For a curve

y=f(x):

(i) the asymptotes parallel to thex-axis are found

by equating the coefficient of the highest power

ofxto zero

(ii) the asymptotes parallel to they-axis are found

by equating the coefficient of the highest power

ofyto zero

With the above example y=

x+ 2

x+ 1

, rearranging

gives:

y(x+1)=x+ 2

i.e. yx+y−x− 2 = 0 (1)

and x(y−1)+y− 2 = 0