Answer: We can start by using the trigonometric identity:
tan(A + B) = (tan A + tan B) / (1 - tan A * tan B)
To find tan A and tan B, we can use the given information:
tan A = 2/3
sin B = sqrt(5)/5
To find tan B, we can use the relationship between sin, cos and tan:
tan B = sin B / cos B
Since sin² B + cos² B = 1, we know that:
cos B = sqrt(1 - sin² B) = sqrt(1 - (5/5)²) = sqrt(1 - (5/25)) = sqrt(20/25) = 2/sqrt(5)
So,
tan B = sin B / cos B = sqrt(5)/5 / 2/sqrt(5) = sqrt(5) / (2*sqrt(5)) = sqrt(5) / 10
Now, we can substitute these values back into the original identity:
tan(A + B) = (2/3 + sqrt(5)/10) / (1 - (2/3)(sqrt(5)/10))
Simplifying this expression gives us:
tan(A + B) = (2/3 + sqrt(5)/10) / (7/15)
And that's the final answer.
Explanation: