190 Higher Engineering Mathematics
(c)arccos
(
−
√
3
2
)
≡cos−^1
(
−
√
3
2
)
= 150 ◦
=
5 π
6
rador2.6180rad
(d)arccosec(
√
2 )=arcsin
(
1
√
2
)
≡sin−^1
(
1
√
2
)
= 45 ◦
=
π
4
rador0.7854rad
Problem 7. Evaluate (in radians), correct to
3 decimal places: sin−^10. 30 +cos−^10 .65.
sin−^10. 30 = 17. 4576 ◦= 0 .3047rad
cos−^10. 65 = 49. 4584 ◦= 0 .8632rad
Hence sin−^10. 30 +cos−^10. 65
= 0. 3047 + 0. 8632 =1.168, correct to 3 decimal places.
Now try the following exercise
Exercise 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
2
or− 1 .5708rad
]
- cos−^10. 5
[π
3
or 1.0472rad
]
- tan−^11
[π
4
or 0.7854rad
]
- cot−^1 2[ 0 .4636rad]
- cosec−^12 .5[0.4115rad]
- sec−^11 .5[0.8411rad]
- sin−^1
(
1
√
2
) [
π
4
or 0.7854rad
]
- Evaluatex, correct to 3 decimal places:
x=sin−^1
1
3
+cos−^1
4
5
−tan−^1
8
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+ 1
is 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.