Question

Evaluate cosA/2 given cosA=-1/3 and tanA >0

Answer 1

`\bold{cos(A)/(2) = -(1)/(\sqrt3)}`

Explanation:

Given that:

`cosA=-\frac{1}3`

and

`tanA > 0`

To find:

`cos(A)/(2) = ?`

Solution:

First of all,we have cos value as negative and tan value as positive.

It is possible in the 3rd quadrant only.

`(A)/(2)` will lie in the 2nd quadrant so
`cos(A)/(2)` will be negative again.

Because Cosine is positive in 1st and 4th quadrant.

Formula:

`cos2\theta =2cos^2(\theta) - 1`

Here
`\theta = (A)/(2)`

`cosA =2cos^2((A)/(2)) - 1\\\Rightarrow 2cos^2((A)/(2)) =cosA+1\\\Rightarrow 2cos^2((A)/(2)) =-\frac{1}3+1\\\Rightarrow 2cos^2((A)/(2)) =\frac{2}3\\\Rightarrow cos((A)/(2)) = \pm (1)/(\sqrt3)`

But as we have discussed,
`cos(A)/(2)` will be negative.

So, answer is:

`\bold{cos(A)/(2) = -(1)/(\sqrt3)}`