Trigonometry Notes 29 April 2011
Solving trigonometric equations where there is a coefficient in front of x
Example 1. Solve for . (no restrictions on infinitely many solutions)
is the angle whose tangent value is . There are infinitely many such angles, so we start with those
the first time around the unit circle. A tangent value of 1 or -1 implies adjacent and opposite side
lengths are the same, so we are looking at a or
reference angle. Tangent is positive in quadrants I
& III and negative in quadrants II & IV, so we want the angles in quadrants II and IV with
reference
angles. This yields:
and
Remember, though, that we are looking for ALL such angles, and they will all be coterminal with these
two, so we add , where is an integer, to each solution to get
and
We want to solve for rather than , so we now have to divide both sides of each equation by 3 to get
This is a perfectly acceptable solution. Notice however, that the angles
and
differ exactly by , so
rather than writing
and
, we could simply write the single expression
, which would cover all of the same angles as the two former expressions. This yields a
much nicer, simpler solution:
.
Example 2. Solve for . (no restrictions on infinitely many solutions)
(
) √
is the angle whose cosine value is √
, and again, there are infinitely many such angles. The first
time around the unit circle, they are the
reference angles in quadrants I & IV, where cosine is positive.
