If csc(−
3
4π
)=x, then the value of x is:
- A3
2
- B−
3
2
- C2
- D−2
Solution & Step-by-step Explanation
We are given:
x=csc(−
3
4π
)
Using the property csc(−θ)=−cscθ:
x=−csc(
3
4π
)
We can rewrite
3
4π
as π+
3
π
:
x=−csc(π+
3
π
)
Since csc(π+θ)=−cscθ (as it lies in the third quadrant where cosecant is negative):
x=−(−csc
3
π
)=csc
3
π
We know that csc
3
π
=
sin
3
π
1
=
2
3
1
=
3
2
.
Thus, x=
3
2
.
x=csc(−
3
4π
)
Using the property csc(−θ)=−cscθ:
x=−csc(
3
4π
)
We can rewrite
3
4π
as π+
3
π
:
x=−csc(π+
3
π
)
Since csc(π+θ)=−cscθ (as it lies in the third quadrant where cosecant is negative):
x=−(−csc
3
π
)=csc
3
π
We know that csc
3
π
=
sin
3
π
1
=
2
3
1
=
3
2
.
Thus, x=
3
2
.