find the exact value of tan 2β if sin β=5/13 (β in quadrant II)

tanβ = -5/12

tan2β = 2tanβ/(1-tan^2β)
= 2(-5/12)/(1-25/144)
= -120/119

To find the exact value of tan 2β, we need to use the given information that sin β = 5/13 and the fact that β is in quadrant II.

First, let's find the value of cos β using Pythagorean Identity: cos² β + sin² β = 1.

Given that sin β = 5/13, we can calculate cos β as follows:

cos β = √(1 - sin² β) = √(1 - (5/13)²) = √(1 - 25/169) = √(144/169) = 12/13

Since β is in quadrant II, where sine is positive and cosine is negative, we conclude that cos β = -12/13.

Now, we can use the double-angle identity for tangent:

tan 2β = (2 * tan β) / (1 - tan² β)

Since β is in quadrant II, tan β = sin β / cos β = (5/13) / (-12/13) = -(5/12).

Substituting the value of tan β into the double-angle identity:

tan 2β = (2 * (-(5/12))) / (1 - (-(5/12))²)

Now, let's simplify the expression further:

tan 2β = (-(10/12)) / (1 - (25/144))

tan 2β = (-(10/12)) / (1 - 25/144)

tan 2β = (-(10/12)) / (144/144 - 25/144)

tan 2β = (-(10/12)) / (119/144)

Finally, we can simplify the expression:

tan 2β = -(10/12) * (144/119)

tan 2β = -(120/119)

Therefore, the exact value of tan 2β, when sin β = 5/13 (β in quadrant II), is -(120/119).