Download Equations Involving Inverse Trigonometric Functions

Equations Involving Inverse Trigonometric Functions
In this section, we use the inverse relationships listed below to solve equations:
For
if and only if
For
and
if and only if
For
and
if and only if
and
(
EXAMPLE 1: Solve the equation
) for x in the interval [
(
Solution: First, multiply both sides by 4:
Since
,
]
)
, and we can use the inverse cosine relationship:
(
)
.
(
Finally, divide by 3:
(
EXAMPLE 2: Solve the equation
)
) for x in the interval
Solution: First, subtract 2 from both sides and multiply by −1:
(
)
(
)
(
)
(
)
, so we can use the inverse tangent relationship:
(
(
Finally, subtract π from both sides:
EXAMPLE 3: Solve the equation
Solution: First, divide both sides by 3:
(
)
(
for x.
)
(
Now use the inverse sine relationship:
Finally, multiply by 2 and add 1:
)
√
√
)
√
)
EXAMPLE 4: Find the exact solutions:
Solution: Add
to both sides:
(
Using the inverse cosine relationship:
Let
. Then
) with
, and
.
√
By the Pythagorean identity relating sine and cosine,
√
Using the addition identity for sine, we can rewrite the equation as
(
)
which implies that
√
Thus, the equation becomes
(
Squaring both sides:
√
√
√
√ √
)
Since squaring both sides of the equation may introduce extraneous solutions, we check both solutions:
First, check
Next, check
so
.
(
)
(
You can verify this using a graphing calculator:
)
is a solution.
(
)
so
is not a solution