Let α = tan⁻¹3/4 and β = sin⁻¹5/13. Find sin(α + β), cos(α + β) and tan(α + β)…

Step 1. The triangles. For $\alpha$: $\sin\alpha = \dfrac{3}{5}$ and $\cos\alpha = \dfrac{4}{5}$. For $\beta$: $\sin\beta = \dfrac{5}{13}$ and $\cos\beta = \dfrac{12}{13}$.

Step 2. The sine. $\sin(\alpha + \beta) = \dfrac{3}{5} \times \dfrac{12}{13} + \dfrac{4}{5} \times \dfrac{5}{13} = \dfrac{56}{65} = 0.8615$.

Step 3. The cosine. $\cos(\alpha + \beta) = \dfrac{4}{5} \times \dfrac{12}{13} - \dfrac{3}{5} \times \dfrac{5}{13} = \dfrac{33}{65} = 0.5077$.

Step 4. The tangent. $\dfrac{56/65}{33/65} = \dfrac{56}{33} = 1.697$.