If sin(α + β) = 1 And sin(α – β) = 1/2, where 0 ≤ α,β ≤ π/2, then find the values of tan(α + 2β) And tan(2α + β). ​

Question

If sin(α + β) = 1 And sin(α – β) = 1/2, where 0 ≤ α,β ≤ \(\frac{π}2\), then find the values of tan(α + 2β) And tan(2α + β).

Answer 1

Given sin(α + β) = 1 And sin(α – β) = \(\frac{1}2\) 

⇒ α + β = 90° …(1) And α - β = 30° …(2) 

Adding(1) And(2), 

⇒ 2α = 120° 

∴ α = 60° 

Subtracting(2) from(1), 

⇒ 2β = 60° 

∴ β = 30° 

Then, 

∴ tan(α + 2β) = tan(60° + 2 × 30°) = tan 120° = -√3 

And tan(2α + β) = tan(2 × 60° + 30°) = tan 150° = -(\(\frac{1}{\sqrt3}\))