Final answer:
The exact value of tan(θ + φ) is 7/4; found using the angle sum identity for tangent after calculating the cosines of θ and φ from their sines considering they are in the first quadrant and constitute Pythagorean triples.
Step-by-step explanation:
To find the exact value of tan(θ + φ), given that sin θ = 4/5 and sin φ = 5/13, with both angles between 0 and π/2 and being Pythagorean triples, we will use trigonometric identities and Pythagorean relationships.
First, let's find the cosines of the angles using the Pythagorean theorem. Since sin θ = 4/5, we can determine that cos θ = √(1 - sin2 θ) = √(1 - (4/5)2) = √(1 - 16/25) = 3/5, knowing that θ is in the first quadrant where cosine is positive. Similarly, since sin φ = 5/13, cos φ = √(1 - sin2 φ) = √(1 - (5/13)2) = 12/13.
Now we will use the angle sum identity for tangent, which is tan(θ + φ) = (tan θ + tan φ)/(1 - tan θ tan φ).
We know tan θ = sin θ/cos θ = (4/5)/(3/5) = 4/3 and tan φ = sin φ/cos φ = (5/13)/(12/13) = 5/12.
Plugging these into the identity gives us
tan(θ + φ) = (4/3 + 5/12)/(1 - (4/3 * 5/12))
= (16/12 + 5/12)/(1 - 20/36)
= 21/12 / (16/36)
= (21/12) * (36/16)
= 7/4.
The exact value of tan(θ + φ) is 7/4.