Simple Trigonometric Equations

 

 You may be asked to solve an equation such as

The main things to remember are:

• Find in which quadrants the solution has the correct sign
• Alter the range to suit any multiple angles
• Find the principal value and the associated angles
• Find solutions for the multiple angle, then divide the solutions.
  

 Each rotation can be divided into 4 quadrants, and the sin, cos and tan functions are positive or negative depending on which quadrant. I remember for each particular quadrant which functions are positive with this simple diagram:

 

  One teacher at my college remembers this by the phrase "All Students Take Cocaine", but this is optional! This diagram simply means that all functions are positive in the 1^st quadrant (0° ® 90° ), then the sin, tan and cos are positive in the 2^nd, 3^rd and 4^th quadrants respectively.

 Once the initial or "principal" value is known, we can find the associated angles. As shown below, the value in the 2^nd quadrant is 180- q , the 3^rd is 180+q , and the 4^th is 360- q (or the equivalent in radians).

 

In our example, we know that the principal solution to 2x is (equivalent of 60° ). We alter the range to accommodate the multiple angle, i.e. because we are dealing with 2x, we double the range:

 This means that to find solutions for 2x, we look in 1 positive revolution, and 1 negative revolution:

The sin function is +ve in the 1^st and 2^nd quadrants, so for 0 to 2p radians we get the solutions

In the other direction, from 0 to - 2p radians, the two solutions are at

 Since these 4 solutions are for sin 2x, we now divide them by two to find the solutions for x: