TRIGONOMETRIC EQUATIONS TRIGONOMETRIC EQUATIONS
Some equations which involve trigonometric functions of the unknown may be readily solved by using simple algebraic ideas (as Equation 1 below), while others may be impossible to solve exactly, only approximately (e.g., Equation 2 below):
EXAMPLE 1:
Find all solutions of the equation 
.
Solution:
We can
graphically visualize all the angles u which satisfy the equation
by noticing that 
is the y-coordinate of the point where
the terminal side of the angle u (in standard position) intersects
the unit circle (see Figure 1):
We can see that there are two angles in 
that satisfy
the equation: 
and 
. Since
the period of the sine
function is 
, it follows that all solutions of the original
equation are:
EXERCISE 1
Find all solutions of the equation 
.
Solution.
EXERCISE 2
Find all solutions of the equation 
that
lie in the interval .
Solution.
EXERCISE 3
Find all solutions of the equation 
in the
interval .
Solution.
EXERCISE 4
Solve the equation 
. Restrict
solutions to the interval 
.
Solution.
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Luis Valdez-Sanchez