Basic Trigonometric Equations:
When asked to solve 2x - 1 = 0, we can easily get 2x = 1 and x = d 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 = d
But this is only part of the answer.
eqgraoh1

If we look at the graph of sinθ from 0 to 2π,
we will remember that there are actually
TWO values of θ for which the sinθ = d.

These values are at:
m

or at 30º and  150º.

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

We could arrive at these "other" solutions by adding a multiple of 2π to θ.

h

where n is an integer in [0,∞).

eqgraph2
Most equations, however, limit the answers to trigonometric equations to the domain
[0, 2π] or [0º, 360º].  (Always read the question carefully to determine the given domain.)


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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.

ex1     g

Solution: First, solve for sin x.
        h

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

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

 

a

k

 

ex2     ;

Solution:   First, solve for tan x.
                           '

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

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

 

f

l

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ti84c
For help with trigonometric equations on
your calculator,
click here.
MathBits Calculator Pages


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