Final answer:
To find the exact value of tan 2θ, use the double angle identity for tangent. Given that sin θ = 3/4 and angle θ is in Quadrant I, you can find the value of tan 2θ using the Pythagorean identity and substitution. The exact value of tan 2θ in simplest radical form is -21√7 / 7.
Step-by-step explanation:
To find the exact value of tan 2θ, first we need to find the value of θ. Given that sin θ = 3/4 and angle θ is in Quadrant I, we can use the Pythagorean identity to find the value of cos θ: cos θ = √(1 - sin^2 θ) = √(1 - (3/4)^2) = √(1 - 9/16) = √(16/16 - 9/16) = √(7/16) = √7/4.
Now we can use the double angle identity for tangent: tan 2θ = (2tan θ) / (1 - tan^2 θ). We already know that tan θ = sin θ / cos θ = (3/4) / (√7/4) = 3/√7.
Substituting this value into the double angle identity, we get: tan 2θ = (2(3/√7)) / (1 - (3/√7)^2) = (6/√7) / (1 - 9/7) = (6/√7) / (-2/7) = -21/√7 = -21√7 / 7.