Question

If cosA= -4/5 and A is in the second quadrent, then cos2A= ?

Answer 1

Cos 2 A =
`(7)/(25)` for the given.

Explanation:

Given:

`\cos A=-(4)/(5)`

We know the formula,

`\cos ^(2) A+\sin ^(2) A=1`

`\sin ^(2) A=1-\cos ^(2) A`

`\sin ^(2) A=1-\left(-(4)/(5)\right)^(2)=1-(16)/(25)=(25-16)/(25)=(9)/(25)`

Taking square root, we get

`\sin A=\sqrt{(9)/(25)}=\pm (3)/(5)`

Hence,

`\sin A=+(3)/(5) \text { and } \sin A=-(3)/(5)`

Given as ‘A’ is in second quadrant, so sine value is positive. Therefore,

`\sin A=+(3)/(5)`

The formula for Cos 2 A is given as

`\cos 2 A=\cos ^(2) A-\sin ^(2) A`

Substitute the values, we get

`\cos 2 A=\left(-(4)/(5)\right)^(2)-\left(+(3)/(5)\right)^(2)`

`\cos 2 A=(16)/(25)-(9)/(25)=(16-9)/(25)=(7)/(25)`