Question

Question #dcf23

Answer 1

`theta=60˚,109.4712˚`

Explanation:

We have the equation

`6cos^2theta-costheta=1`

Notice that this is similar to a quadratic equation--we have a variable expression being squared, a variable expression to the first power, and a constant. We should try to solve this as we normally would a quadratic first and then bring the trigonometry into play.

To make this more approachable, a good (although not necessary method) is to let `u=costheta`. Then, we have the equation

`6u^2-u-1=0`

To factor this quadratic, find a pair of factors of `-6` whose sum is `-1`: the numbers `-3` and `2`.

`6u^2-3u+2u-1=0`

`3u(2u-1)+1(2u-1)=0`

`(3u+1)(2u-1)=0`

`{(3u+1=0),(2u-1=0):}`

`{(u=-1/3),(u=1/2):}`

Now, since `u=costheta`,

`{(costheta=-1/3),(costheta=1/2):}`

Solving just for the first equation:

`costheta=-1/3`

Using the inverse cosine function,

`theta=arccos(-1/3)`

Plugging this into a calculator yields `theta=109.4712˚` (approximately).

The second equation is a commonly known value:

`costheta=1/2`

`theta=60˚`