1. Trigonometric Equations with Multiple Angles

2. The diagram shows where the various ratios are positive
Note also that values repeat every 360o
90°
180 - S A
180°
0°,360° C
180 + T
360 -
270°

3. Solve cos 3x = – 1/√2   for values of x such that 0 < x < π
First of all consider the range of values
If 0 < x < π then 0 < 3x < 3π
(540o)
Remember 3π = 3 × 180o = 540o
Work in degrees then change back to radians later
Now find BASE angle
cos -1 (1/√2 ) = 45o
π/4 A S T C 
180 –
Since cosine negative 
3x = 135o , 225o
495o
180 +
Now change back to radians using multiples of π/4
3x = 3π/4 , 5π/4 , 11π/4
So x = 3π/12 , 5π/12 , 11π/12

4. Solve sin 2x = 1/2   for values of x such that 0 < x < 2π
First of all consider the range of values
If 0 < x < 2π then 0 < 2x < 4π
(720o)
Work in degrees then change back to radians later
Now find BASE angle
Remember 4π = 4 × 180o = 720o
sin -1 (1/2 ) = 30o
π/6 A S T C  
180 –
Since sine positive
2x = 30o , 150o
390o , 510o
Now change back to radians using multiples of π/6
2x = π/6 , 5π/6 , 13π/6 ,17π/6
So x = π/12 , 5π/12 , 13π/12 , 17π/12

5. Solve tan 4x = √3   for values of x such that 0 < x < π
First of all consider the range of values
If 0 < x < π then 0 < 4x < 4π
(720o)
Remember 4π = 4 × 180o = 720o
Now find BASE angle
Work in degrees then change back to radians later
tan -1 (√3 ) = 60o
π/3 A S T C 
Since tangent positive 
180 +
4x = 60o , 240o
420o , 600o
Now change back to radians using multiples of π/3
4x = π/3 , 4π/3 , 7π/3 ,10π/3
So x = π/12 , π/3 , 7π/12 , 5π/6

6. Solve 2sin (2x – π/6) = √3   2x – π/6 = 60o , 120o
for values of x such that 0 < x < 2π
First of all consider the range of values
If 0 < x < 2π then 0 < 2x – π/6 < 4π – π/6)
(690o)
Remember 2π – π/6 = 2 × 180o – 30o = 690o
Now find BASE angle
Work in degrees then change back to radians later A S T C  
sin -1 (√3/2 ) = 60o
π/3
180 -
2x – π/6 = 60o , 120o
420o , 480o
Now change back to radians using multiples of π/3
2x – π/6 = π/3 , 2π/3 , 7π/3 ,8π/3
So x = π/4 , 5π/12 , 5π/6 , 17π/12
2x = 3π/6 , 5π/6 , 15π/6 ,17π/6