Solving Limits Problem

[april19]

I need to solve x in the following function so as x is approaching 3.
I think my problem right now is more a trig. problem because I can't solve it.

f(x)=5-2sin(x-3) When x=3, f(x)=5

What is x when f(x)=5.5 and f(x)=4.5?

Here's what I got so far.

5-2sin(x-3)= 5.5
2sin(x-3)=0.5
sin(x-3)=0.25

I cheated and saw the answer has arcsin in it. I can't remember how to solve this.
Any help from anybody?
Thanks.

 

[royhaas]

Apply $\displaystyle arcsin$ to both sides of your last equation.

 

[BigGlenntheHeavy]

$\displaystyle f(x) \ = \ 5-2sin(x-3).$

$\displaystyle \lim_{x\to3}f(x) \ = \ 5-2sin(3-3) \ = \ 5-2sin(0) \ = \ 5-2(0) \ = \ 5.$

$\displaystyle f(x) \ = \ 5.5 \ = \ 5-2sin(x-3), \ solve \ for \ x.$

$\displaystyle -2sin(x-3) \ = \ .5$

$\displaystyle sin(x-3) \ = \ \frac{-1}{4}$

$\displaystyle (x-3) \ = \ arcsin(-1/4) \ \dot= \ -.25268$

$\displaystyle Hence, \ x \ \dot= \ 2.74732$

$\displaystyle Now, \ you \ should \ be \ able \ to \ do \ the \ next \ one.$

 

[april19]

Great! Thanks!!!
Wrecked my brain on this one. Can't believe it was that simple to solve it.