Question

Evaluate the Value of 2sinpi/12

Answer 1

4
(
√
6
−
√
2
)
Explanation:
We want to find replacement angles for
π
12
that will produce exact values
These must come from :
π
6
,
π
3
,
π
4
⇒
sin
(
π
12
)
=
sin
(
π
3
−
π
4
)
Using the appropriate
Addition formula
∣
∣
∣
∣
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
a
a
sin
(
A
±
B
)
=
sin
A
cos
B
±
cos
A
sin
B
a
a
∣
∣
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
⇒
sin
(
π
3
−
π
4
)
=
sin
(
π
3
)
cos
(
π
4
)
−
cos
(
π
3
)
sin
(
π
4
)
Extract
exact values from triangles

sin
(
π
3
)
=
√
3
2
,
sin
(
π
4
)
=
1
√
2

and
cos
(
π
3
)
=
1
2
,
cos
(
π
4
)
=
1
√
2

now substitute into the right side of the expansion.
=
√
3
2
×
1
√
2
−
1
2
×
1
√
2
=
√
3
2
√
2
−
1
2
√
2
=
√
3
−
1
2
√
2
and rationalising the denominator
gives
(
√
3
−
1
)
×
√
2
2
√
2
×
√
2
=
√
6
−
√
2
4

Answer 2

Hope this will help you..