Answer:
60° and 300°
Step-by-step 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:
- [tex]sinx^2 + cosx^2 = 1[/tex]
You'll see why this formula is useful soon.
Work:
[tex]4cosx+4sinx^2=5[/tex]
- 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 [tex]cosx[/tex] in our equation instead of both [tex]sinx[/tex] and [tex]cosx[/tex] 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 [tex]sinx[/tex].
[tex]sinx^2+cosx^2=1[/tex]
- We are going to isolate [tex]sinx^2[/tex] and then substitute what its equal to into the equation.
[tex]sinx^2=-cosx^2+1[/tex]
- Now Substitute this back into the original equation.
[tex]4cosx+4(-cosx^2+1)=5[/tex]
- Put this into standard form. Then we can factor.
[tex]4cosx-4cosx^2+4=5[/tex]
[tex]-4cosx^2+4cosx+4=5[/tex]
[tex]4cosx^2-4cosx-4=-5[/tex]
[tex]4cosx^2-4cosx+1=0[/tex]
- Let [tex]cosx[/tex] be [tex]u[/tex].
[tex]4u^2-4u+1=0[/tex]
- Look familiar? We could have factored before just the same, but this ensures there will be fewer mistakes while factoring.
[tex](2u-1)^2=0[/tex]
- Now let's change [tex]u[/tex] back to [tex]cosx[/tex]
[tex](2cosx-1)^2=0[/tex]
- Set [tex]2cosx-1[/tex] equal to 0.
[tex]2cosx-1=0[/tex]
[tex]2cosx=1[/tex]
[tex]cosx=1/2[/tex]
- Hopefully if your teacher taught you unit circle properly, you will know that [tex]cosx=1/2[/tex] 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.
[tex](cos^-^1)cosx=1/2(cos^-^1)[/tex]
[tex]x=60[/tex] [tex]x=300[/tex]
