What is the exact value of 75°? Show your work.

Split
75
75
into two angles where the values of the six trigonometric functions are known.
sin
(
30
+
45
)
sin
(
30
+
45
)
Apply the sum of angles identity.
sin
(
30
)
cos
(
45
)
+
cos
(
30
)
sin
(
45
)
sin
(
30
)
cos
(
45
)
+
cos
(
30
)
sin
(
45
)
The exact value of
sin
(
30
)
sin
(
30
)
is
1
2
1
2
.
1
2
cos
(
45
)
+
cos
(
30
)
sin
(
45
)
1
2
cos
(
45
)
+
cos
(
30
)
sin
(
45
)
The exact value of
cos
(
45
)
cos
(
45
)
is
√
2
2
2
2
.
1
2
⋅
√
2
2
+
cos
(
30
)
sin
(
45
)
1
2
⋅
2
2
+
cos
(
30
)
sin
(
45
)
The exact value of
cos
(
30
)
cos
(
30
)
is
√
3
2
3
2
.
1
2
⋅
√
2
2
+
√
3
2
sin
(
45
)
1
2
⋅
2
2
+
3
2
sin
(
45
)
The exact value of
sin
(
45
)
sin
(
45
)
is
√
2
2
2
2
.
1
2
⋅
√
2
2
+
√
3
2
⋅
√
2
2
1
2
⋅
2
2
+
3
2
⋅
2
2
Simplify
1
2
⋅
√
2
2
+
√
3
2
⋅
√
2
2
1
2
⋅
2
2
+
3
2
⋅
2
2
.
Tap for more steps...
√
2
+
√
6
4
2
+
6
4
The result can be shown in multiple forms.
Exact Form:
√
2
+
√
6
4
2
+
6
4
Decimal Form:
0.96592582
…