If θ and ϕ are acute angles such that sin θ = 1/2 and cos ϕ = 1/

1.	If θ and ϕ are acute angles such that sin θ = 1/2 and cos ϕ = 1/3 , than θ + ϕ lies in(a) \(\bigg]\frac{π}{3},\fracπ2\bigg[\)(b) \(\bigg]\frac{2π}{3},\frac{5π}3\bigg[\)(c) \(\bigg]\frac{π}{2},\frac{2π}3\bigg[\)(d) \(\bigg]\frac{5π}{6},π\bigg[\)
Answer» (c) \(\bigg]\frac{π}{2},\frac{2π}3\bigg[\) sin θ = \(\frac12\) ⇒ sin θ = sin \(\fracπ6\) ⇒ θ = \(\fracπ6\) (∵ θ and φ are acute angles lying in the first quadrant) …(i) Now cos ϕ = \(\frac13\) ⇒ 0 < cos ϕ < \(\frac12\) ⇒ cos \(\fracπ2\) < cos ϕ < cos \(\fracπ3\) ⇒ \(\fracπ2\) < ϕ < \(\fracπ3\) …(ii) ∴ From (i) and (ii) \(\fracπ2\) + \(\fracπ6\) < θ + ϕ < \(\fracπ3\) + \(\fracπ6\) ⇒ \(\fracπ2\) < θ + ϕ < \(\frac{2π}3\) ⇒ θ + ϕ lies in the open interval \(\bigg]\frac{π}{2},\frac{2π}3\bigg[\). Hence (c) is the correct option.

If θ and ϕ are acute angles such that sin θ = 1/2 and cos ϕ = 1/3 , than θ + ϕ lies in(a) \(\bigg]\frac{π}{3},\fracπ2\bigg[\)(b) \(\bigg]\frac{2π}{3},\frac{5π}3\bigg[\)(c) \(\bigg]\frac{π}{2},\frac{2π}3\bigg[\)(d) \(\bigg]\frac{5π}{6},π\bigg[\)

Answer»

(c) \(\bigg]\frac{π}{2},\frac{2π}3\bigg[\)

sin θ = \(\frac12\) ⇒ sin θ = sin \(\fracπ6\) ⇒ θ = \(\fracπ6\)

(∵ θ and φ are acute angles lying in the first quadrant) …(i)

Now cos ϕ = \(\frac13\) ⇒ 0 < cos ϕ < \(\frac12\) ⇒ cos \(\fracπ2\) < cos ϕ < cos \(\fracπ3\) ⇒ \(\fracπ2\) < ϕ < \(\fracπ3\) …(ii)

∴ From (i) and (ii) \(\fracπ2\) + \(\fracπ6\) < θ + ϕ < \(\fracπ3\) + \(\fracπ6\)

⇒ \(\fracπ2\) < θ + ϕ < \(\frac{2π}3\) ⇒ θ + ϕ lies in the open interval \(\bigg]\frac{π}{2},\frac{2π}3\bigg[\).

Hence (c) is the correct option.