102 Higher Engineering Mathematics
(a) cot 17.49 ◦ =
1
tan 17.49 ◦ = 3.1735
(b) cot 163 ◦ 52 =
1
tan 163 ◦ 52 =
1
tan 163
52 ◦
60
= −3.4570
Problem 12. Evaluate, correct to 4 significant
figures:
(a) sin 1.481 (b) cos(3π/5) (c) tan 2.93
(a) sin 1.481 means the sine of 1.481 radians. Hence
a calculator needs to be on the radian function.
Hence sin 1.481 = 0.9960
(b) cos(3π/5) = cos 1.884955 · · · =−0.3090
(c) tan 2.93 = −0.2148
Problem 13. Evaluate, correct to 4 decimal
places:
(a) secant 5.37 (b) cosecant π/4
(c) cotangent π/24
(a) Again, with no degrees sign, it is assumed that
5.37 means 5.37 radians.
Hence sec 5.37 =
1
cos 5.37
= 1.6361
(b) cosec (π/4) =
1
sin(π/4)
=
1
sin 0.785398 ...
= 1.4142
(c) cot(5π/24) =
1
tan(5π/24)
=
1
tan 0.654498 ...
= 1.3032
Problem 14. Find, in degrees, the acute angle
sin −1 0.4128 correct to 2 decimal places.
sin
−1 0.4128 means ‘the angle whose
sine is 0.4128’
Using a calculator:
1. Press shift 2. Press sin 3. Enter 0.4128
4. Press ) 5. Press = The answer 24.380848……
is displayed
Hence,
sin
−1 0.4128 = 24.38
◦
Problem 15. Find the acute angle cos −1 0.2437 in
degrees and minutes
cos
−1 0.2437 means ‘the angle whose
cosine is 0.2437’
Using a calculator:
1. Press shift 2. Press cos 3. Enter 0.2437
4. Press ) 5. Press = The answer 75.894979…
is displayed
6. Press ◦ ”’ and 75 ◦ 53 41.93 is displayed
Hence, cos
−1 0.2437 = 75.89
◦
= 77
◦ 54
correct to the nearest minute.
Problem 16. Find the acute angle tan −1 7.4523 in
degrees and minutes
tan
−1 7.4523 means ‘the angle whose
tangent is 7.4523’
Using a calculator:
1. Press shift 2. Press tan 3. Enter 7.4523
4. Press ) 5. Press = The answer 82.357318…
is displayed
6. Press ◦ ”’ and 82 ◦ 21 26.35 is displayed
Hence, tan
−1 7.4523 = 82.36
◦
= 82
◦ 21
correct to the nearest minute.
Problem 17. Determine the acute angles:
(a) sec −1 2.3164 (b) cosec −1 1.1784
(c) cot −1 2.1273
(a) sec
−1 2.3164 = cos
−1
1
2.3164
= cos
−1 0.4317 ...
= 64.42
◦ or 64
◦ 25
or 1.124 radians
(b) cosec −1 1.1784 = sin
−1
1
1.1784
= sin
−1 0.8486 ...
= 58.06
◦ or 58
◦ 4
or 1.013 radians
(a) cot 17.49 ◦ =
1
tan 17.49 ◦ = 3.1735
(b) cot 163 ◦ 52 =
1
tan 163 ◦ 52 =
1
tan 163
52 ◦
60
= −3.4570
Problem 12. Evaluate, correct to 4 significant
figures:
(a) sin 1.481 (b) cos(3π/5) (c) tan 2.93
(a) sin 1.481 means the sine of 1.481 radians. Hence
a calculator needs to be on the radian function.
Hence sin 1.481 = 0.9960
(b) cos(3π/5) = cos 1.884955 · · · =−0.3090
(c) tan 2.93 = −0.2148
Problem 13. Evaluate, correct to 4 decimal
places:
(a) secant 5.37 (b) cosecant π/4
(c) cotangent π/24
(a) Again, with no degrees sign, it is assumed that
5.37 means 5.37 radians.
Hence sec 5.37 =
1
cos 5.37
= 1.6361
(b) cosec (π/4) =
1
sin(π/4)
=
1
sin 0.785398 ...
= 1.4142
(c) cot(5π/24) =
1
tan(5π/24)
=
1
tan 0.654498 ...
= 1.3032
Problem 14. Find, in degrees, the acute angle
sin −1 0.4128 correct to 2 decimal places.
sin
−1 0.4128 means ‘the angle whose
sine is 0.4128’
Using a calculator:
1. Press shift 2. Press sin 3. Enter 0.4128
4. Press ) 5. Press = The answer 24.380848……
is displayed
Hence,
sin
−1 0.4128 = 24.38
◦
Problem 15. Find the acute angle cos −1 0.2437 in
degrees and minutes
cos
−1 0.2437 means ‘the angle whose
cosine is 0.2437’
Using a calculator:
1. Press shift 2. Press cos 3. Enter 0.2437
4. Press ) 5. Press = The answer 75.894979…
is displayed
6. Press ◦ ”’ and 75 ◦ 53 41.93 is displayed
Hence, cos
−1 0.2437 = 75.89
◦
= 77
◦ 54
correct to the nearest minute.
Problem 16. Find the acute angle tan −1 7.4523 in
degrees and minutes
tan
−1 7.4523 means ‘the angle whose
tangent is 7.4523’
Using a calculator:
1. Press shift 2. Press tan 3. Enter 7.4523
4. Press ) 5. Press = The answer 82.357318…
is displayed
6. Press ◦ ”’ and 82 ◦ 21 26.35 is displayed
Hence, tan
−1 7.4523 = 82.36
◦
= 82
◦ 21
correct to the nearest minute.
Problem 17. Determine the acute angles:
(a) sec −1 2.3164 (b) cosec −1 1.1784
(c) cot −1 2.1273
(a) sec
−1 2.3164 = cos
−1
1
2.3164
= cos
−1 0.4317 ...
= 64.42
◦ or 64
◦ 25
or 1.124 radians
(b) cosec −1 1.1784 = sin
−1
1
1.1784
= sin
−1 0.8486 ...
= 58.06
◦ or 58
◦ 4
or 1.013 radians