Question

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

Answer 1

Answer:

$\bold{cos\dfrac{A}{2} = -\dfrac{1}{\sqrt3}}$

Step-by-step explanation:

Given that:

$cosA=-\dfrac{1}3$

and

$tanA > 0$

To find:

$cos\dfrac{A}{2} = ?$

Solution:

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

It is possible in the 3rd quadrant only.

$\dfrac{A}{2}$ will lie in the 2nd quadrant so $cos\dfrac{A}{2}$ will be negative again.

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

Formula:

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

Here $\theta = \frac{A}{2}$

$cosA =2cos^2(\dfrac{A}{2}) - 1\\\Rightarrow 2cos^2(\dfrac{A}{2}) =cosA+1\\\Rightarrow 2cos^2(\dfrac{A}{2}) =-\dfrac{1}3+1\\\Rightarrow 2cos^2(\dfrac{A}{2}) =\dfrac{2}3\\\Rightarrow cos(\dfrac{A}{2}) = \pm \dfrac{1}{\sqrt3}$

But as we have discussed, $cos\dfrac{A}{2}$ will be negative.

So, answer is:

$\bold{cos\dfrac{A}{2} = -\dfrac{1}{\sqrt3}}$