To calculate tan θ, we can use the relationship between sine (sin θ) and tangent (tan θ):
tan θ = sin θ / cos θ
To find the value of tan θ, we need to determine the value of cos θ. We know that sin θ = 6/9, and we can use the Pythagorean identity to find cos θ:
cos^2 θ + sin^2 θ = 1
Since sin θ = 6/9, we can substitute it in the equation:
cos^2 θ + (6/9)^2 = 1
Simplifying:
cos^2 θ + 36/81 = 1
cos^2 θ = 1 - 36/81
cos^2 θ = 45/81
Now, find the value of cos θ:
cos θ = √(45/81) ≈ √(5/9) ≈ √(5) / 3
Now, we can calculate tan θ:
tan θ = sin θ / cos θ
tan θ = (6/9) / (√(5) / 3)
tan θ = (6/9) * (3 / √(5))
tan θ = 18 / (9 * √(5))
tan θ = 2 / √(5)
To rationalize the denominator, multiply both the numerator and denominator by √(5):
tan θ = (2 / √(5)) * (√(5) / √(5))
tan θ = 2√(5) / 5
So, the value of tan θ is 2√(5) / 5.