If sin theta = 5/13 and Oterminates on the interval [0, pi 2] find the exact value of tan 2theta
a.12/5
b. 120/119
c. 5/12
d.119/120
sinθ = 5/13, so in QI,
tanθ = 5/12
tan2θ = 2tanθ / (1 - tan^2 θ)
so plug and chug
To find the exact value of tan(2θ), we can use the double-angle identity for tangent:
tan(2θ) = 2tan(θ) / (1 - tan²(θ))
First, let's find the value of tan(θ) using the given information. We are given that sin(θ) = 5/13. We can use the Pythagorean identity to find cos(θ):
cos²(θ) = 1 - sin²(θ)
cos²(θ) = 1 - (5/13)²
cos²(θ) = 1 - 25/169
cos²(θ) = 169/169 - 25/169
cos²(θ) = 144/169
Taking the square root of both sides, we get:
cos(θ) = ±12/13
Since θ terminates on the interval [0, π/2], sin(θ) is positive, so we take the positive value for cos(θ):
cos(θ) = 12/13
Now, we can calculate tan(θ) using the definition of tangent:
tan(θ) = sin(θ) / cos(θ)
tan(θ) = (5/13) / (12/13)
tan(θ) = 5/12
Now that we have the value of tan(θ), we can use the double-angle identity to find tan(2θ):
tan(2θ) = 2tan(θ) / (1 - tan²(θ))
tan(2θ) = 2 * (5/12) / (1 - (5/12)²)
tan(2θ) = (10/12) / (1 - 25/144)
tan(2θ) = (10/12) / (144/144 - 25/144)
tan(2θ) = (10/12) / (119/144)
tan(2θ) = (10/12) * (144/119)
tan(2θ) = 120/119
Therefore, the exact value of tan(2θ) is b. 120/119.