If \(\cos α=\frac{2}{3}\) and \(\sinβ=\frac{1}{4}\), then what is the value of cos (α - β)?

Concept use:

cos (α - β) = cosα cosβ + sinα sinβ 

Calculations:

\(\cos α=\frac{2}{3}\) = Base/Hypotenuse 

By Pythagoras theorem  H2 = P2 + B2

⇒ 3² - 2² = P² 

⇒ P = √5

sin α = √5/3

\(\sinβ=\frac{1}{4}\) = Perpendicular/Hypotenuse 

By Pythagoras theorem  H2 = P2 + B2

⇒ 42 - 12 = B2 

⇒ B = √15

cosβ = √15/4

cos (α - β) = cosα cosβ + sinα sinβ = 2/3 × √15/4 + √5/3 × 1/4 = \(\frac{2\sqrt{15}+\sqrt5}{12}\)

Hence, The Correct Answer is \(\frac{2\sqrt{15}+\sqrt5}{12}\)