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SOLVING TRIGONOMETRIC EQUATIONS - PART 2
Example 8:
for x on the interval
Solve
degrees.
. Express your answers in EXACT
Since the equation is "like a quadratic equation" of the form
the Square Root Property
That is,
to help us solve for x .
and
This indicates that we must find solutions for x, given
for x on the interval
a. Solve
radians.
, we will use
and
. Express your answers in EXACT
Step 1:
We are going to use the concept of Inverse Trigonometric Functions
to solve for the angle x using the calculator.
The numeric value is positive !
Using the calculator in degree mode, we find
Step 2:
Reference Angle for angle x equals 45o.
Step 3:
First, we are going to find the quadrants in which the solutions must
be found. We use All Students Take Calculus and the fact that
once cos x is isolated on one side, its numeric value is
positive. See Step 1 above.
It is best to draw a picture showing the solution interval
and the quadrants in which we have to find the
angles.
.
Finally, we are going to use the Reference Angle from Step 2. We
need to find two angles located in QI and QIV that have a Reference
Angle of 45o. We know that Reference Angles in these quadrants
are calculated as follows:
is a first-quadrant angle then
is a fourth-quadrant angle then
,
which implies that
Therefore, the solutions for
for x on the interval
b. Solve
degrees.
are
. Express your answers in EXACT
Step 1:
We are going to use the concept of Inverse Trigonometric Functions
to solve for the angle x using the calculator.
The numeric value is negative !
Using the calculator in degree mode, we find
NOTE: It's easier to use degrees in the solution process! Then,
for your final answer, change the solutions to radians!
Step 2:
Reference Angle for angle x equals 180o - 135o = 45o .
Please note that the range of the inverse cosine is limited to the
. That is why the calculator gives us 135o.
interval
Step 3:
First, we are going to find the quadrants in which the solutions must
be found. We use All Students Take Calculus and the fact that
once cos x is isolated on one side, its numeric value is
negative. See Step 1 above.
It is best to draw a picture showing the solution interval
and the quadrants in which we have to find the
angles.
Finally, we are going to use the Reference Angle from Step 2. We
need to find two angles located in QII and QIII that have a Reference
Angle of 45o. We know that Reference Angles in these quadrants
are calculated as follows:
is a second-quadrant angle then
,
which implies that
is a third-quadrant angle the
, which
implies that
Therefore, the solutions for
Then the solutions for
are
are 45o , 135o, 225o , and 315o .