Question

If sin (α + β)= 1 and sin (α - β)= ½, where 0 ≤ α , β ≤/2 , then find the value of tan (α + 2β) and tan (2α + β).

Answer 1

If sin (α + β)= 1 and sin (α - β)= ½, where 0 ≤ α , β ≤/2 , then find the value of tan (α + 2β) and tan (2α + β).

sin (α + β)= 1
α + β=90 degrees ...(1)

sin (α - β)= ½

α - β= 30 degrees ...(2)

(1)+(2) gives
2α=120
α=60 degrees

(1) gives,
60+β=90
β=30 degrees

Now,
tan (α + 2β)=tan(60+2(30))= tan(60+60)=tan 120 = tan (90+30)= -cot30=$-\sqrt{3}$
tan (2α + β)=tan(2(60)+30)=tan(120+30)=tan150=tan(180-30)=-tan30=$\frac{-1}{\sqrt{3}}$