156 Higher Engineering Mathematics
In Problems 4 to 6, solve for θ in the range
0
◦
≤ θ ≤ 360
◦ .
4. sec θ = 2
[ 6 0 ◦ , 300 ◦ ]
5. cot θ = 0.6
[ 5 9 ◦ , 239 ◦ ]
6. cosec θ = 1.5
[ 4 1 .81 ◦ , 138.19 ◦ ]
In Problems 7 to 9, solve for x in the range
−180
◦
≤ x ≤ 180
◦ .
7. sec x = −1.5
[ ±131.81 ◦ ]
8. cot x = 1.2
[ 3 9 .81
◦
, −140.19
◦ ]
9. cosec x = −2
[ −30
◦
, −150
◦ ]
In Problem 10 and 11, solve for θ in the range
0 ◦ ≤ θ ≤ 360 ◦ .
10. 3 sin θ = 2 cosθ
[33.69 ◦ , 213.69 ◦ ]
11. 5 cos θ = −sin θ
[101.31
◦
, 281.31
◦ ]
15.5 Worked problems (ii) on
trigonometric equations
Problem 10. Solve 2 −4 cos
2 A = 0 for values of
A in the range 0 ◦ < A < 360 ◦ .
2 − 4 cos 2 A = 0, from which cos 2 A =
2
4 = 0.5000
Hence cos A =
√
(0.5000) =±0.7071 and
A = cos
−1
(±0.7071).
Cosine is positive in quadrants one and four and negative in quadrants two and three. Thus in this case there
are four solutions, one in each quadrant (see Fig. 15.6).
The acute angle cos −1 0.7071 =45 ◦ . Hence,
A = 45
◦
, 135
◦
, 225
◦ or 315
◦
Problem 11. Solve
1
2 cot 2 y = 1.3 for
0 ◦ < y < 360 ◦ .
1
2 cot 2 y = 1.3, from which, cot 2 y = 2(1.3) = 2.6
Hence cot y =
√
2.6 =±1.6125, and y = cot −1
(±1.6125). There are four solutions, one in each
quadrant. The acute angle cot −1 1.6125 =31.81 ◦ .
Hence y = 31.81 ◦ , 148.19 ◦ , 211.81 ◦ or 328.19 ◦ .
1.0
y
A8
y 5 cos A
0
0.7071
1358
458
1808
3158 3608
2258
20.7071
21.0
(a)
(b)
458
458
458
T
C
A
S
458
1808
3608
2708
908
0
Figure 15.6
Now try the following exercise
Exercise 67 Further problems on
trigonometric equations
In Problems 1 to 3 solve the equations for angles
between 0 ◦ and 360 ◦ .
1. 5 sin
2 y = 3
y = 50.77 ◦ , 129.23 ◦ ,
230.77 ◦ or 309.23 ◦
2. cos 2 θ = 0.25
[θ = 60 ◦ , 120 ◦ , 240 ◦ or 300 ◦ ]
3. tan 2 x = 3
[θ = 60 ◦ , 120 ◦ , 240 ◦ or 300 ◦ ]
4. 5 + 3 cosec 2 D = 8
[D = 90 ◦ or 270 ◦ ]
5. 2 cot 2 θ = 5
θ = 32.32 ◦ , 147.68 ◦ ,
212.32 ◦ or 327.68 ◦
In Problems 4 to 6, solve for θ in the range
0
◦
≤ θ ≤ 360
◦ .
4. sec θ = 2
[ 6 0 ◦ , 300 ◦ ]
5. cot θ = 0.6
[ 5 9 ◦ , 239 ◦ ]
6. cosec θ = 1.5
[ 4 1 .81 ◦ , 138.19 ◦ ]
In Problems 7 to 9, solve for x in the range
−180
◦
≤ x ≤ 180
◦ .
7. sec x = −1.5
[ ±131.81 ◦ ]
8. cot x = 1.2
[ 3 9 .81
◦
, −140.19
◦ ]
9. cosec x = −2
[ −30
◦
, −150
◦ ]
In Problem 10 and 11, solve for θ in the range
0 ◦ ≤ θ ≤ 360 ◦ .
10. 3 sin θ = 2 cosθ
[33.69 ◦ , 213.69 ◦ ]
11. 5 cos θ = −sin θ
[101.31
◦
, 281.31
◦ ]
15.5 Worked problems (ii) on
trigonometric equations
Problem 10. Solve 2 −4 cos
2 A = 0 for values of
A in the range 0 ◦ < A < 360 ◦ .
2 − 4 cos 2 A = 0, from which cos 2 A =
2
4 = 0.5000
Hence cos A =
√
(0.5000) =±0.7071 and
A = cos
−1
(±0.7071).
Cosine is positive in quadrants one and four and negative in quadrants two and three. Thus in this case there
are four solutions, one in each quadrant (see Fig. 15.6).
The acute angle cos −1 0.7071 =45 ◦ . Hence,
A = 45
◦
, 135
◦
, 225
◦ or 315
◦
Problem 11. Solve
1
2 cot 2 y = 1.3 for
0 ◦ < y < 360 ◦ .
1
2 cot 2 y = 1.3, from which, cot 2 y = 2(1.3) = 2.6
Hence cot y =
√
2.6 =±1.6125, and y = cot −1
(±1.6125). There are four solutions, one in each
quadrant. The acute angle cot −1 1.6125 =31.81 ◦ .
Hence y = 31.81 ◦ , 148.19 ◦ , 211.81 ◦ or 328.19 ◦ .
1.0
y
A8
y 5 cos A
0
0.7071
1358
458
1808
3158 3608
2258
20.7071
21.0
(a)
(b)
458
458
458
T
C
A
S
458
1808
3608
2708
908
0
Figure 15.6
Now try the following exercise
Exercise 67 Further problems on
trigonometric equations
In Problems 1 to 3 solve the equations for angles
between 0 ◦ and 360 ◦ .
1. 5 sin
2 y = 3
y = 50.77 ◦ , 129.23 ◦ ,
230.77 ◦ or 309.23 ◦
2. cos 2 θ = 0.25
[θ = 60 ◦ , 120 ◦ , 240 ◦ or 300 ◦ ]
3. tan 2 x = 3
[θ = 60 ◦ , 120 ◦ , 240 ◦ or 300 ◦ ]
4. 5 + 3 cosec 2 D = 8
[D = 90 ◦ or 270 ◦ ]
5. 2 cot 2 θ = 5
θ = 32.32 ◦ , 147.68 ◦ ,
212.32 ◦ or 327.68 ◦