Find the value of (sin75
∘
+sin15
∘
).
- A2
6
- B2
3
- C6
- D3
Solution & Step-by-step Explanation
We can use the trigonometric sum-to-product formula:
sinC+sinD=2sin(
2
C+D
)cos(
2
C−D
)
Substituting C=75
∘
and D=15
∘
:
sin75
∘
+sin15
∘
=2sin(
2
75
∘
+15
∘
)cos(
2
75
∘
−15
∘
)
sin75
∘
+sin15
∘
=2sin(45
∘
)cos(30
∘
)
We know the standard values:
sin(45
∘
)=
2
1
,cos(30
∘
)=
2
3
Substituting these values back:
sin75
∘
+sin15
∘
=2×
2
1
×
2
3
=
2
3
Multiplying the numerator and denominator by
2
to match the choices:
2
×
2
3
×
2
=
2
6
sinC+sinD=2sin(
2
C+D
)cos(
2
C−D
)
Substituting C=75
∘
and D=15
∘
:
sin75
∘
+sin15
∘
=2sin(
2
75
∘
+15
∘
)cos(
2
75
∘
−15
∘
)
sin75
∘
+sin15
∘
=2sin(45
∘
)cos(30
∘
)
We know the standard values:
sin(45
∘
)=
2
1
,cos(30
∘
)=
2
3
Substituting these values back:
sin75
∘
+sin15
∘
=2×
2
1
×
2
3
=
2
3
Multiplying the numerator and denominator by
2
to match the choices:
2
×
2
3
×
2
=
2
6