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Solve for x: tan(sin(x)) + 1 = 2

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1. Subtract 1 from both sides to isolate the term tan⁡(sin⁡(�))tan(sin(x)): tan⁡(sin⁡(�))=2−1tan(sin(x))=2−1 tan⁡(sin⁡(�))=1tan(sin(x))=1

2. We know that tan⁡(�)=1tan(θ)=1 when �=�4θ=4π​ or �=5�4θ=45π​ (plus any integer multiple of �π).

3. However, sin⁡(�)sin(x) can take any value between -1 and 1, so �x can be any angle whose sine is either �44π​ or 5�445π​.

Thus, the solutions for �x are:

�=arcsin⁡(�4)+2��x=arcsin(4π​)+2nπ �=arcsin⁡(5�4)+2��x=arcsin(45π​)+2nπ

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Some background knowledge we used was algebra. We recognized that there were extra parentheses. Which reminds us of f(g(x)). This is said as f of g of x. Meaning g(x) is inserted into the x of the f(x). We also knew to subtract 1 from each side to have the equation equal to something and to be isolated. So, tan(sin(x))= 1 and we would need to manipulate the problem to get tan(sin(x)) to equal 1.

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After the term the was isolated, then you take the inverse. You then use the period to find the set of all solutions.