Determine the value of Tan2x when
Sinx = -12/13 and 3pi/2 < x < 2pi.
Tan2x is Sin2x/cos2x
but Sin2x is 2sinxcosx and
Cos2x=1-2sin^2x
you know sinx, and you can find cos x from pyth theorm (sides of thriangle are 13, 12,5)
To determine the value of Tan2x, we can use the trigonometric identity:
Tan2x = (2tanx) / (1 - tan²x)
Given that Sinx = -12/13 and 3π/2 < x < 2π, we need to find the value of Cosx in order to calculate Tanx.
Since Sinx = -12/13, we can use the Pythagorean identity to find Cosx. The Pythagorean identity is:
Sin²x + Cos²x = 1
Substituting the given value of Sinx into the identity, we get:
(-12/13)² + Cos²x = 1
144/169 + Cos²x = 1
Cos²x = 1 - 144/169
Cos²x = (169 - 144)/169
Cos²x = 25/169
Since 3π/2 < x < 2π, we know that Cosx < 0 and Tanx = Sinx / Cosx. Therefore, we have:
Cosx = -√(25/169) (since Cosx < 0)
Now, we can calculate Tanx:
Tanx = Sinx / Cosx
= (-12/13) / (-√(25/169))
= (-12/13) / (-5/13)
Simplifying the expression gives:
Tanx = 12 / 5
Finally, we can substitute the value of Tanx into the Tan2x identity to find Tan2x:
Tan2x = (2tanx) / (1 - tan²x)
= (2 * (12/5)) / (1 - (12/5)²)
= (24/5) / (1 - (144/25))
= (24/5) / (25/25 - 144/25)
= (24/5) / (-119/25)
= -24/119
Therefore, the value of Tan2x is -24/119.