190 Higher Engineering Mathematics
(c) arccos
−
√
3
2
≡ cos
−1
−
√
3
2
= 150
◦
=
5π
6
rad or 2.6180 rad
(d) arccosec(
√
2) = arcsin
1
√
2
≡ sin
−1
1
√
2
= 45
◦
=
π
4
rad or 0.7854 rad
Problem 7. Evaluate (in radians), correct to
3 decimal places: sin −1 0.30 + cos −1 0.65.
sin
−1 0.30 = 17.4576
◦
= 0.3047 rad
cos
−1 0.65 = 49.4584
◦
= 0.8632 rad
Hence sin −1 0.30 + cos −1 0.65
= 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.
1. f (x) = x + 1
[f −1 (x) = x − 1]
2. f (x) = 5x − 1
f −1 (x) =
1
5 (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 −1 0.5
π
3
or 1.0472 rad
7. tan −1 1
π
4
or 0.7854 rad
8. cot −1 2
[ 0 .4636 rad]
9. cosec −1 2.5
[ 0 .4115 rad]
10. sec −1 1.5
[ 0 .8411 rad]
11. sin −1
1
√
2
π
4
or 0.7854 rad
12. Evaluate x, correct to 3 decimal places:
x = sin
−1 1
3
+ cos
−1 4
5
− tan
−1 8
9
[0.257]
13. Evaluate y, correct to 4 significant figures:
y = 3 sec
−1
√
2 − 4 cosec
−1
√
2
+ 5 cot
−1 2
[1.533]
18.7 Asymptotes
If a table of values for the function y =
x + 2
x + 1
is drawn
up for various values of x and then y plotted against x,
the graph would be as shown in Fig. 18.32. The straight
lines AB, i.e. x =−1, and CD, 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 asymptote can be considered as a tangent to the curve at
infinity.
Asymptotes parallel to the x- and y-axes
There is a simple rule which enables asymptotes parallel to the x- and y-axis to be determined. For a curve
y = f (x):
(i) the asymptotes parallel to the x-axis are found by
equating the coefficient of the highest power of x
to zero.
(ii) the asymptotes parallel to the y-axis are found by
equating the coefficient of the highest power of y
to zero.
(c) arccos
−
√
3
2
≡ cos
−1
−
√
3
2
= 150
◦
=
5π
6
rad or 2.6180 rad
(d) arccosec(
√
2) = arcsin
1
√
2
≡ sin
−1
1
√
2
= 45
◦
=
π
4
rad or 0.7854 rad
Problem 7. Evaluate (in radians), correct to
3 decimal places: sin −1 0.30 + cos −1 0.65.
sin
−1 0.30 = 17.4576
◦
= 0.3047 rad
cos
−1 0.65 = 49.4584
◦
= 0.8632 rad
Hence sin −1 0.30 + cos −1 0.65
= 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.
1. f (x) = x + 1
[f −1 (x) = x − 1]
2. f (x) = 5x − 1
f −1 (x) =
1
5 (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 −1 0.5
π
3
or 1.0472 rad
7. tan −1 1
π
4
or 0.7854 rad
8. cot −1 2
[ 0 .4636 rad]
9. cosec −1 2.5
[ 0 .4115 rad]
10. sec −1 1.5
[ 0 .8411 rad]
11. sin −1
1
√
2
π
4
or 0.7854 rad
12. Evaluate x, correct to 3 decimal places:
x = sin
−1 1
3
+ cos
−1 4
5
− tan
−1 8
9
[0.257]
13. Evaluate y, correct to 4 significant figures:
y = 3 sec
−1
√
2 − 4 cosec
−1
√
2
+ 5 cot
−1 2
[1.533]
18.7 Asymptotes
If a table of values for the function y =
x + 2
x + 1
is drawn
up for various values of x and then y plotted against x,
the graph would be as shown in Fig. 18.32. The straight
lines AB, i.e. x =−1, and CD, 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 asymptote can be considered as a tangent to the curve at
infinity.
Asymptotes parallel to the x- and y-axes
There is a simple rule which enables asymptotes parallel to the x- and y-axis to be determined. For a curve
y = f (x):
(i) the asymptotes parallel to the x-axis are found by
equating the coefficient of the highest power of x
to zero.
(ii) the asymptotes parallel to the y-axis are found by
equating the coefficient of the highest power of y
to zero.