Question

Find the exact value of tan(θ + φ) given that sin θ = 4/5 and sin φ = 5/13and that θ and φ are between 0 and π/2. Note these angles are Pythagorean Triples.

Answer 1

$\sin\theta=\dfrac45\implies\cos\theta=\dfrac35$
$\sin\varphi=\dfrac5{13}\implies\cos\varphi=\dfrac{12}{13}$


$\tan\theta=\dfrac{\sin\theta}{\cos\theta}=\dfrac45$
$\tan\varphi=\dfrac{\sin\varphi}{\cos\varphi}=\dfrac5{12}$

$\tan(\theta+\varphi)=\dfrac{\tan\theta+\tan\varphi}{1-\tan\theta\tan\varphi}=\dfrac{63}{16}$