Solving Trig Equations 1
90
Sine
All
2nd
1st
180
The cast diagram shows you the quadrants in which
the named trig functions are positive. In the example
below we want the quadrants where sine is negative.
Sine is positive in the 1st and 2nd quadrants, so it
must be negative in the 3rd and 4th quadrants.
0
Tan
3rd
Cosine
270
4th
Solve the following trig equations giving angles in the range 0  x  360 to 1 decimal place
Equation
Principal Value
(value from calculator)
Other Quadrant
Cast Diagram
17.5
Example:
sin x  0.3
1. tan x  0.2
2. cos x  0.6
3. sin x  0.15
4. tan x  0.76
5. cos x  0.43

17.5
3rd
All solutions in the range
17.5
197.5 ,342.5

2
Solving Trig Equations 2
 1.571
S
A
0
  3.142
T
C
3
 4.712
2
Solve the following trig equations giving angles in the range 0  x  2
Equation
Example:
tan x  2
Principal Value
(value from calculator)
Other Quadrant
x  1.11c
3rd
Cast Diagram
1.11
1.11
1. sin x  0.82
2. tan x  0.94
3. cos x  0.49
4. cos x  0.77
5. sin x  0.23
All solutions in the range
x  1.11c , 4.25c
Solving Trig Equations 3

Solve the following trig equations giving angles in the range 0  x  360 , giving your answers to one decimal place
Equation
Example
7  6 cos 2 x  5sin x
Working out
7  6 1  sin 2 x   5sin x
6sin 2 x  5sin x  1  0
 3sin x  1 2sin x  1  0
1
1
sin x  ,sin x 
3
2
1.
7  2 cos x  8sin 2 x
2.
2sin x  3cos x  0
3.
cos  3x  15   0.8
(For this one you will
need to do the stages
in a different order:
principal value first,
then cast diagram,
then working out)
Principal
Value(s)
(from
calculator)
19.5 ,30
Cast Diagram (or diagrams)
30
19.5
All solutions in the
range
x  19.5 ,30 ,150 ,160.5
Solving Trig Equations 4
Solve the following trig equations giving angles in the range 0  x  2 , giving your answers to 2 decimal places
Equation
Example
4cos x  3sin x  0
Working Out
2.
2 cos 2 x  sin x  1
3.
tan  2 x  0.1c   1
(Remember to do the
stages in a different
order for this one.)
4.
tan 2 x  tan x  2  0
Cast Diagram (or diagrams)
All solutions in the
range
3sin x  4 cos x
3 tan x  4
tan x 
1.
2sin x  cos x  0
Principal
Value(s)
4
3
0.93c
0.93
x  0.93c , 4.07c
Trig Equations Extension A
Solve the following trig equations for the range given:
Question
1.
sin x  0.61
180  x  180
2.
cos x  0.39
0  x  720
3.
tan x  1.61
  x  
4.
sin x  0.71
180  x  360
5.
cos x  0.23
720  x  0
6.
cos x  0.68
2  x  0
7.
tan x  3
2  x  2
8.
sin x  0.25
360  x  360
9.
tan x  1.61
0  x  4
10.
cos x  0.3
  x  3
Working out
Solution
Trig Equations Extension B
Solve the following trig equations for the range given:
Question
1.
sin 2 x  0.61
180  x  180
2.
x

cos   20   0.39
2



0  x  720
3.
sin x  2cos x
  x  
4.
3sin x  2cos x  0
180  x  360
5.
4sin 2 x  2  2cos x
720  x  0
6.
2 tan 2 x  5tan x  3
2  x  0
7.
3tan  x  0.2c   2
2  x  2
8.
13  12cos2 x  7sin x  0
360  x  360
Working out
Solution
Solutions to Trig Equation Sheets: 1, 2 and Extension A
Sheet 1:
1.
2.
3.
4.
5.
11.3 ,191.3
53.1 ,306.9
8.6 ,171.4
142.8 ,322.8
115.5 ,244.5
Sheet 2:
1.
2.
3.
4.
5.
0.96c ,2.18c
2.39c ,5.53c
2.08c ,4.20c
0.69c ,5.59c
6.05c ,3.37c
Extension A:
1. 37.6 ,142.4
2. 67.0 ,293.0 ,427.0 ,653.0
3. 1.01c ,2.13c
4. 134.8 , 45.2 ,225.2 ,314.8
5. 643.3 , 436.7 , 283.3 , 76.7
6. 3.96c , 2.32c
7. 4.39c , 1.25c ,1.89c ,5.03c
8. 345.5 , 194.5 ,14.5 ,165.5
9. 1.01c ,4.16c ,7.30c ,10.44c
10. 1.27c ,1.27c ,5.02c ,7.55c
Solutions to Trig Equation Sheets: 3, 4 and Extension B
Sheet 3:
1.
2.
3.
75.5 ,120 , 240 , 284.5
56.3 , 236.3
7.3 ,102.7 ,127.3 , 222.7 , 247.3 ,342.7
Sheet 4:
1.
2.
3.
4.
2.68c ,5.82c
1.57c ,3.67c ,5.76c
0.44c , 2.01c ,3.58c ,5.16c
1.11c , 2.36c , 4.25c ,5.50c
Extension B:
1.
2.
3.
4.
5.
6.
7.
8.
161.2 , 108.8 ,18.8 ,71.2
94.1 ,545.9
1.10c , 2.03c
146.3 ,33.7 , 213.7
600 , 480 , 360 , 240 , 120 ,0
5.82c , 4.39c , 2.68c , 1.25c
3.93c , 0.79c , 2.35c ,5.50c
165.5 , 160.5 , 19.5 , 14.5 ,194.5 ,199.5 ,340.5 ,345.5