Cos^2x=1/2
Can you please solve this?

Take the inverse cosine of both sides of the equation to extract
x
x
from inside the cosine.
2
x
=
arccos
(
1
2
)
2
x
=
arccos
(
1
2
)
The exact value of
arccos
(
1
2
)
arccos
(
1
2
)
is
π
3
π
3
.
2
x
=
π
3
2
x
=
π
3
Divide each term by
2
2
and simplify.
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x
=
π
6
x
=
π
6
The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from
2
π
2
π
to find the solution in the fourth quadrant.
2
x
=
2
π
−
π
3
2
x
=
2
π
-
π
3
Simplify the expression to find the second solution.
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x
=
5
π
6
x
=
5
π
6
Find the period.
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π
π
The period of the
cos
(
2
x
)
cos
(
2
x
)
function is
π
π
so values will repeat every
π
π
radians in both directions.
x
=
π
6
+
π
n
,
5
π
6
+
π
n
x
=
π
6
+
π
n
,
5
π
6
+
π
n
, for any integer
n

The decimal expansion would be .27 repeated