It is given that sin a = 2/7 and a is an acute angle.
We use the relation (sin x)^2 + (cos x)^2 = 1
(cos a)^2 + (sin a)^2 = 1
=> (cos a)^2 + (2/7)^2 = 1
=> (cos a)^2 = 1 - 4/49
=> (cos a)^2 = 45/49
as the angle a is acute cos a is positive.
=> cos a = sqrt 45/7
The value of cos a = (sqrt 45)/7
Since a is an acute angle, that means that a<`pi/2` , therefore we'll calculate the value of cosine function for an angle located within the 1st quadrant. In the 1st quadrant, all the values of trigonometric functions are positive.
We'll use Pythagorean identity to compute cos a:
`cos^2 a = 1 - sin^2 a`
`cos a = +sqrt(1 - sin^2 a)`
cos a = `+sqrt(1 - 4/49)`
cos a = `+sqrt (45/49)`
cos a = `+(3sqrt5)/7`
Therefore, the requested value of cosine of the acute angle a is:
cos a = `+(3sqrt5)/7` .