Question

If `sin B = -12/13` then what is `cos 2B` ?

Answer 1

`sin"B"=12/13`
`"B"=sin^-1(-12/13)`

Explanation:

Once you have worked out the value of `"B"`, you simply have to type `cos(2"B")` in your calculator to get the answer. However, this will give you the principle value of `2"B"`. To find out the value of `2"B"` in the third quadrant, do `360-cos(2"B")`.

The number of significant figures/decimal places is up to you.
Another tip is that if you want a more accurate answer you can type `cos(2xxsin^-1(-12/13))` into your calculator. This will help you to avoid rounding errors. Then, you can do `360-"ANS"` (instead of rounding).

Answer 2

`cos 2B = -119/169`

Explanation:

Given:

`sin B = -12/13`

Then:

`cos 2B = 2 cos^2 B - 1`

`color(white)(cos 2B) = 2(1-sin^2 B) - 1`

`color(white)(cos 2B) = 1-2 sin^2 B`

`color(white)(cos 2B) = 1-2 (-12/13)^2`

`color(white)(cos 2B) = 1-288/169`

`color(white)(cos 2B) = (169-288)/169`

`color(white)(cos 2B) = -119/169`