Introduction to trigonometry 103
(c) cot −1 2.1273 = tan −1
1
2.1273
= tan
−1 0.4700 ...
= 25.18
◦ or 25
◦ 11
or 0.439 radians
Problem 18. Evaluate the following expression,
correct to 4 significant figures:
4 sec32
◦ 10
− 2 cot 15
◦ 19
3 cosec 63 ◦ 8 tan 14 ◦ 57
By calculator:
sec 32
◦ 10
= 1.1813, cot 15
◦ 19
= 3.6512
cosec 63
◦ 8
= 1.1210, tan 14
◦ 57
= 0.2670
Hence
4 sec32 ◦ 10 − 2 cot 15 ◦ 19
3 cosec 63 ◦ 8 tan 14 ◦ 57
=
4(1.1813) − 2(3.6512)
3(1.1210)(0.2670)
=
4.7252 − 7.3024
0.8979
=
−2.5772
0.8979
= −2.870,
correct to 4 significant figures.
Problem 19. Evaluate correct to 4 decimal places:
(a) sec(−115 ◦ ) (b) cosec (−95 ◦ 47 )
(a) Positive angles are considered by convention to be
anticlockwise and negative angles as clockwise.
Hence −115 ◦ is actually the same as 245 ◦ (i.e.
360 ◦ − 115 ◦ )
Hence sec(−115
◦
) = sec 245
◦
=
1
cos 245 ◦
= −2.3662
(b) cosec (−95 ◦ 47 ) =
1
sin
−95
47 ◦
60
= −1.0051
Problem 20. In triangle E FG in Fig. 11.11,
calculate angle G.
2.30
8.71
F
E
G
Figure 11.11
With reference to ∠G, the two sides of the triangle
given are the opposite side E F and the hypotenuse
E G; hence, sine is used,
i.e.
sin G =
2.30
8.71
= 0.26406429 ...
from which,
G = sin −1 0.26406429 ...
i.e.
G = 15.311360 ...
Hence,
∠G = 15.31
◦ or 15
◦ 19
Now try the following exercise
Exercise 46 Further problems on
evaluating trigonometric ratios
In Problems 1 to 8, evaluate correct to 4 decimal
places:
1. (a) sine 27 ◦ (b) sine 172.41 ◦
(c) sine 302 ◦ 52 (a) 0.4540 (b) 0.1321
(c) −0.8399
2. (a) cosine 124 ◦ (b) cosine 21.46 ◦
(c) cosine 284 ◦ 10
(a) −0.5592 (b) 0.9307
(c) 0.2447
3. (a) tangent 145 ◦ (b) tangent 310.59 ◦
(c) tangent 49 ◦ 16
(a) −0.7002 (b) −1.1671
(c) 1.1612
4. (a) secant 73 ◦ (b) secant 286.45 ◦
(c) secant 155 ◦ 41
(a) 3.4203 (b) 3.5313
(c) −1.0974
(c) cot −1 2.1273 = tan −1
1
2.1273
= tan
−1 0.4700 ...
= 25.18
◦ or 25
◦ 11
or 0.439 radians
Problem 18. Evaluate the following expression,
correct to 4 significant figures:
4 sec32
◦ 10
− 2 cot 15
◦ 19
3 cosec 63 ◦ 8 tan 14 ◦ 57
By calculator:
sec 32
◦ 10
= 1.1813, cot 15
◦ 19
= 3.6512
cosec 63
◦ 8
= 1.1210, tan 14
◦ 57
= 0.2670
Hence
4 sec32 ◦ 10 − 2 cot 15 ◦ 19
3 cosec 63 ◦ 8 tan 14 ◦ 57
=
4(1.1813) − 2(3.6512)
3(1.1210)(0.2670)
=
4.7252 − 7.3024
0.8979
=
−2.5772
0.8979
= −2.870,
correct to 4 significant figures.
Problem 19. Evaluate correct to 4 decimal places:
(a) sec(−115 ◦ ) (b) cosec (−95 ◦ 47 )
(a) Positive angles are considered by convention to be
anticlockwise and negative angles as clockwise.
Hence −115 ◦ is actually the same as 245 ◦ (i.e.
360 ◦ − 115 ◦ )
Hence sec(−115
◦
) = sec 245
◦
=
1
cos 245 ◦
= −2.3662
(b) cosec (−95 ◦ 47 ) =
1
sin
−95
47 ◦
60
= −1.0051
Problem 20. In triangle E FG in Fig. 11.11,
calculate angle G.
2.30
8.71
F
E
G
Figure 11.11
With reference to ∠G, the two sides of the triangle
given are the opposite side E F and the hypotenuse
E G; hence, sine is used,
i.e.
sin G =
2.30
8.71
= 0.26406429 ...
from which,
G = sin −1 0.26406429 ...
i.e.
G = 15.311360 ...
Hence,
∠G = 15.31
◦ or 15
◦ 19
Now try the following exercise
Exercise 46 Further problems on
evaluating trigonometric ratios
In Problems 1 to 8, evaluate correct to 4 decimal
places:
1. (a) sine 27 ◦ (b) sine 172.41 ◦
(c) sine 302 ◦ 52 (a) 0.4540 (b) 0.1321
(c) −0.8399
2. (a) cosine 124 ◦ (b) cosine 21.46 ◦
(c) cosine 284 ◦ 10
(a) −0.5592 (b) 0.9307
(c) 0.2447
3. (a) tangent 145 ◦ (b) tangent 310.59 ◦
(c) tangent 49 ◦ 16
(a) −0.7002 (b) −1.1671
(c) 1.1612
4. (a) secant 73 ◦ (b) secant 286.45 ◦
(c) secant 155 ◦ 41
(a) 3.4203 (b) 3.5313
(c) −1.0974