190 Higher Engineering Mathematics
(c)arccos(
−√
3
2)
≡cos−^1(
−√
3
2)
= 150 ◦=5 π
6rador2.6180rad(d)arccosec(√
2 )=arcsin(
1
√
2)≡sin−^1(
1
√
2)
= 45 ◦=π
4rador0.7854radProblem 7. Evaluate (in radians), correct to
3 decimal places: sin−^10. 30 +cos−^10 .65.sin−^10. 30 = 17. 4576 ◦= 0 .3047radcos−^10. 65 = 49. 4584 ◦= 0 .8632radHence sin−^10. 30 +cos−^10. 65
= 0. 3047 + 0. 8632 =1.168, correct to 3 decimal places.Now try the following exerciseExercise 79 Further problems on inverse
functions
Determine the inverse of the functions given in
Problems 1 to 4.- f(x)=x+1[f−^1 (x)=x−1]
- f(x)= 5 x− 1
[
f−^1 (x)=^15 (x+ 1 )]- f(x)=x^3 +1[f−^1 (x)=^3
√
x−1]- f(x)=
1
x+ 2[
f−^1 (x)=1
x− 2]Determine the principal value of the inverse func-
tions in Problems 5 to 11.- sin−^1 (− 1 )
[
−π
2or− 1 .5708rad]- cos−^10. 5
[π
3or 1.0472rad]- tan−^11
[π
4or 0.7854rad]- cot−^1 2[ 0 .4636rad]
- cosec−^12 .5[0.4115rad]
- sec−^11 .5[0.8411rad]
- sin−^1
(
1
√
2) [
π
4or 0.7854rad]- Evaluatex, correct to 3 decimal places:
x=sin−^11
3+cos−^14
5−tan−^18
9
[0.257]- Evaluatey, correct to 4 significant figures:
y=3sec−^1√
2 −4cosec−^1√
2+5cot−^12
[1.533]18.7 Asymptotes
If a table of values for the functiony=x+ 2
x+ 1is drawn
up for various values ofxand thenyplotted againstx,
the graph would be as shown in Fig. 18.32. The straight
linesAB,i.e.x=−1, andCD,i.e.y=1, are known as
asymptotes.
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 paral-
lel 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 ofx
to zero.(ii) the asymptotes parallel to they-axis are found by
equating the coefficient of the highest power ofy
to zero.