Question

4cos+4sin^2=5 find cos​

Answer 1

60° and 300°

Explanation:

Before beginning to answer this question, you'll need a quick formula that will be very helpful trying to answer questions like this one:

• 
  `sinx^2 + cosx^2 = 1`

You'll see why this formula is useful soon.

Work:

`4cosx+4sinx^2=5`

• The way we want to go about answering this question is very similar to what we've learned in the past: factoring. In order to be able to factor though, we need to have only
  `cosx` in our equation instead of both
  `sinx` and
  `cosx` like the above equation has. this is where the formula I showed earlier comes into play.

• Using the equation I showed earlier, we are going to get rid of the
  `sinx`.

`sinx^2+cosx^2=1`

• We are going to isolate
  `sinx^2` and then substitute what its equal to into the equation.

`sinx^2=-cosx^2+1`

• Now Substitute this back into the original equation.

`4cosx+4(-cosx^2+1)=5`

• Put this into standard form. Then we can factor.

`4cosx-4cosx^2+4=5`

`-4cosx^2+4cosx+4=5`

`4cosx^2-4cosx-4=-5`

`4cosx^2-4cosx+1=0`

• Let
  `cosx` be
  `u`.

`4u^2-4u+1=0`

• Look familiar? We could have factored before just the same, but this ensures there will be fewer mistakes while factoring.

`(2u-1)^2=0`

• Now let's change
  `u` back to
  `cosx`

`(2cosx-1)^2=0`

• Set
  `2cosx-1` equal to 0.

`2cosx-1=0`

`2cosx=1`

`cosx=1/2`

• Hopefully if your teacher taught you unit circle properly, you will know that
  `cosx=1/2` is asking for an angle. In this case, I know the angle because of memorization, but sadly the only way to learn this stuff is memorization, so I wont be able to show work for how I got my final answer. I'll have to just show calculator-wise how its done.

`(cos^-^1)cosx=1/2(cos^-^1)`

`x=60`
`x=300`

Answer 2

cos θ = 1/2

Explanation:

4 cos θ + 4 sin² θ = 5

Use Pythagorean identity.

4 cos θ + 4 (1 − cos² θ) = 5

4 cos θ + 4 − 4 cos² θ = 5

0 = 4 cos² θ − 4 cos θ + 1

0 = (2 cos θ − 1)²

0 = 2 cos θ − 1

cos θ = 1/2