Trigonometric Equations
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Basic Trigonometric Equations:

When asked to solve 2x - 1 = 0, we can easily get 2x = 1 and x = as the answer.

When asked to solve 2sinx - 1 = 0, we proceed in a similar manner.  We first look at sinx as being the variable of the equation and solve as we did in the first example.

 

2sinx - 1 = 0

2sinx = 1

sinx =

But, we are only half way to the answer!!!

If we recall the graph of , we will remember that there are actually TWO values of for which the = 1/2.

These values are at:
 

or at 30º and  150º.
 

If we look at the extended graph of , we see that there are many other solutions to this equation = 1/2. 

We could arrive at these "other" solutions by adding a multiple of to .

.

Most equations, however, limit the answers to trigonometric equations to the domain .  (Always read the question carefully to determine the given domain.)

 

Signs and Quadrants:

Solutions of trigonometric equations may also be found by examining the sign of the trig value and determining the proper quadrant(s) for that value.

Example 1:    

Solution:   First, solve for sin x.

                          

Now, sine is negative in Quadrant III and Quadrant IV. 

Also, a sine value of is a reference angle of 45º.   So, consider the reference angle of 45º in quadrants III and IV.

 

 

Example 2:    

Solution:   First, solve for tan x.

                          

Now, tangent is negative in Quadrant II and Quadrant IV. 

Also, a tangent value of is a reference angle of 60 degrees.   So, consider the reference angle of 60º in quadrants II and IV.

 

 

How to use your
TI-83+/84+ graphing calculator  to solve trigonometric equations.
Click calculator.