Question
Find the exact value csc^2(2theta) if theta= pi/12
Answers
GPT 3.5
We know that csc(theta) is the reciprocal of sin(theta). To find the value of csc^2(2theta), we first need to find sin(2theta).
Let's find sin(2theta) using the double-angle formula for sine:
sin(2theta) = 2sin(theta)cos(theta)
Since theta = pi/12, we have:
sin(2(pi/12)) = 2sin(pi/12)cos(pi/12)
Using the values of sin(pi/12) = 1/2√2 and cos(pi/12) = √3/2√2 (found from the unit circle or special triangles), we have:
sin(2(pi/12)) = 2 * (1/2√2) * (√3/2√2)
= (√3/2)
Now, we can find csc^2(2theta):
csc^2(2theta) = (1/sin(2theta))^2
= (1/√3/2)^2
= (2/√3)^2
= 4/3
Therefore, the exact value of csc^2(2theta) is 4/3 when theta = pi/12.
Let's find sin(2theta) using the double-angle formula for sine:
sin(2theta) = 2sin(theta)cos(theta)
Since theta = pi/12, we have:
sin(2(pi/12)) = 2sin(pi/12)cos(pi/12)
Using the values of sin(pi/12) = 1/2√2 and cos(pi/12) = √3/2√2 (found from the unit circle or special triangles), we have:
sin(2(pi/12)) = 2 * (1/2√2) * (√3/2√2)
= (√3/2)
Now, we can find csc^2(2theta):
csc^2(2theta) = (1/sin(2theta))^2
= (1/√3/2)^2
= (2/√3)^2
= 4/3
Therefore, the exact value of csc^2(2theta) is 4/3 when theta = pi/12.