Which is cos2 theta if sin theta = -2/sqrt13 and 180 < theta < 270?
1. -(13/5)
2. -(5/13)
3. 5/13
4. 13/5
3?
correct.
To find the value of cos2(theta) when sin(theta) is given and its value lies in the third quadrant (180 < theta < 270), you can use the Pythagorean identity for sine and cosine:
sin^2(theta) + cos^2(theta) = 1
Given that sin(theta) = -2/sqrt(13), we can substitute the value and solve for cos^2(theta):
(-2/sqrt(13))^2 + cos^2(theta) = 1
4/13 + cos^2(theta) = 1
cos^2(theta) = 1 - 4/13
cos^2(theta) = 13/13 - 4/13
cos^2(theta) = 9/13
Since we are looking for cos2(theta), we can use the identity:
cos2(theta) = 2cos^2(theta) - 1
Substituting the value of cos^2(theta), we get:
cos2(theta) = 2(9/13) - 1
cos2(theta) = 18/13 - 1
cos2(theta) = 18/13 - 13/13
cos2(theta) = 5/13
Therefore, the correct answer is option 3: 5/13.