Introduction to trigonometry 101
values, correct to 4 decimal places, may be checked:
secant 32
◦
=
1
cos 32 ◦ = 1.1792
cosecant 75
◦
=
1
sin 75 ◦ = 1.0353
cotangent 41
◦
=
1
tan 41 ◦ = 1.1504
secant 215.12
◦
=
1
cos 215.12 ◦ = −1.2226
cosecant 321.62
◦
=
1
sin 321.62 ◦ = −1.6106
cotangent 263.59
◦
=
1
tan 263.59 ◦ = 0.1123
If we know the value of a trigonometric ratio and need
to find the angle we use the inverse function on our
calculators.
For example, using shift and sin on our calculator gives
sin −1 (
If, for example, we know the sine of an angle is 0.5 then
the value of the angle is given by:
sin
−1 0.5 = 30
◦
(Check that sin 30
◦
= 0.5)
(Note that sin −1 x does not mean
1
sin x ; also, sin −1 x may
also be written as arcsin x)
Similarly, if cos θ = 0.4371 then
θ = cos −1 0.4371 = 64.08
◦
and if tan A = 3.5984 then A = tan −1 3.5984
= 74.47
◦
each correct to 2 decimal places.
Use your calculator to check the following worked
examples.
Problem 6. Determine, correct to 4 decimal
places, sin 43
◦ 39
sin 43
◦ 39
= sin 43
39
60
◦
= sin 43.65
◦
= 0.6903
This answer can be obtained using the calculator as
follows:
1. Press sin
2. Enter 43
3. Press
◦ ”’
4. Enter 39
5. Press ◦ ”’
6. Press )
7. Press = Answer = 0.6902512….
Problem 7. Determine, correct to 3 decimal
places, 6 cos 62 ◦ 12
6 cos62
◦ 12
= 6 cos62
12 ◦
60
= 6 cos62.20
◦
= 2.798
This answer can be obtained using the calculator as
follows:
1. Enter 6
2. Press cos
3. Enter 62
4. Press ◦ ”’
5. Enter 12
6. Press ◦ ”’
7. Press)
8. Press = Answer = 2.798319….
Problem 8. Evaluate correct to 4 decimal places:
(a) sine 168 ◦ 14 (b) cosine 271.41 ◦
(c) tangent 98 ◦ 4
(a) sine 168 ◦ 14 = sine 168
14
◦
60
= 0.2039
(b) cosine 271.41 ◦ = 0.0246
(c) tangent 98 ◦ 4 = tan 98
4
◦
60
= −7.0558
Problem 9. Evaluate, correct to 4 decimal places:
(a) secant 161 ◦ (b) secant 302 ◦ 29
(a) sec 161 ◦ =
1
cos 161 ◦ = −1.0576
(b) sec 302 ◦ 29 =
1
cos 302 ◦ 29 =
1
cos 302
29 ◦
60
= 1.8620
Problem 10. Evaluate, correct to 4 significant
figures:
(a) cosecant 279.16 ◦ (b) cosecant 49 ◦ 7
(a) cosec 279.16 ◦ =
1
sin 279.16 ◦ = −1.013
(b) cosec 49 ◦ 7 =
1
sin 49 ◦ 7 =
1
sin 49
7 ◦
60
= 1.323
Problem 11. Evaluate, correct to 4 decimal
places:
(a) cotangent 17.49 ◦ (b) cotangent 163 ◦ 52
values, correct to 4 decimal places, may be checked:
secant 32
◦
=
1
cos 32 ◦ = 1.1792
cosecant 75
◦
=
1
sin 75 ◦ = 1.0353
cotangent 41
◦
=
1
tan 41 ◦ = 1.1504
secant 215.12
◦
=
1
cos 215.12 ◦ = −1.2226
cosecant 321.62
◦
=
1
sin 321.62 ◦ = −1.6106
cotangent 263.59
◦
=
1
tan 263.59 ◦ = 0.1123
If we know the value of a trigonometric ratio and need
to find the angle we use the inverse function on our
calculators.
For example, using shift and sin on our calculator gives
sin −1 (
If, for example, we know the sine of an angle is 0.5 then
the value of the angle is given by:
sin
−1 0.5 = 30
◦
(Check that sin 30
◦
= 0.5)
(Note that sin −1 x does not mean
1
sin x ; also, sin −1 x may
also be written as arcsin x)
Similarly, if cos θ = 0.4371 then
θ = cos −1 0.4371 = 64.08
◦
and if tan A = 3.5984 then A = tan −1 3.5984
= 74.47
◦
each correct to 2 decimal places.
Use your calculator to check the following worked
examples.
Problem 6. Determine, correct to 4 decimal
places, sin 43
◦ 39
sin 43
◦ 39
= sin 43
39
60
◦
= sin 43.65
◦
= 0.6903
This answer can be obtained using the calculator as
follows:
1. Press sin
2. Enter 43
3. Press
◦ ”’
4. Enter 39
5. Press ◦ ”’
6. Press )
7. Press = Answer = 0.6902512….
Problem 7. Determine, correct to 3 decimal
places, 6 cos 62 ◦ 12
6 cos62
◦ 12
= 6 cos62
12 ◦
60
= 6 cos62.20
◦
= 2.798
This answer can be obtained using the calculator as
follows:
1. Enter 6
2. Press cos
3. Enter 62
4. Press ◦ ”’
5. Enter 12
6. Press ◦ ”’
7. Press)
8. Press = Answer = 2.798319….
Problem 8. Evaluate correct to 4 decimal places:
(a) sine 168 ◦ 14 (b) cosine 271.41 ◦
(c) tangent 98 ◦ 4
(a) sine 168 ◦ 14 = sine 168
14
◦
60
= 0.2039
(b) cosine 271.41 ◦ = 0.0246
(c) tangent 98 ◦ 4 = tan 98
4
◦
60
= −7.0558
Problem 9. Evaluate, correct to 4 decimal places:
(a) secant 161 ◦ (b) secant 302 ◦ 29
(a) sec 161 ◦ =
1
cos 161 ◦ = −1.0576
(b) sec 302 ◦ 29 =
1
cos 302 ◦ 29 =
1
cos 302
29 ◦
60
= 1.8620
Problem 10. Evaluate, correct to 4 significant
figures:
(a) cosecant 279.16 ◦ (b) cosecant 49 ◦ 7
(a) cosec 279.16 ◦ =
1
sin 279.16 ◦ = −1.013
(b) cosec 49 ◦ 7 =
1
sin 49 ◦ 7 =
1
sin 49
7 ◦
60
= 1.323
Problem 11. Evaluate, correct to 4 decimal
places:
(a) cotangent 17.49 ◦ (b) cotangent 163 ◦ 52