Dethrone
- 716
- 0
In integration, we are allowed to use identities such as $$sinx = \sqrt{1-cos^2x}$$. Why does that work, and why doesn't make a difference in integration? Graphing $$\sqrt{1-cos^2x}$$ is only equal to sinx on certain intervals such as$$ (0, \pi) $$and $$(2\pi, 3\pi)$$. More correctly, shouldn't we use the absolute value of $$\sin\left({x}\right)$$?
$$sin^2x = 1 - cos^2x$$
$$|sinx| = \sqrt{1 = cos^2x}$$
or defined piecewisely = {$$\sin\left({x}\right)$$ in regions where it is above the x-axis, and -$$\sin\left({x}\right)$$ in regions where x is below the x-axis.
Is it possible to even truly isolate "$$\sin\left({x}\right)$$" from
$$sin^2x = 1 - cos^2x$$? It seems as the |$$\sin\left({x}\right)$$| is the closest we can to isolate it.
Sorry if I may seem confusing, but the concept of absolute value still confuses me.
$$sin^2x = 1 - cos^2x$$
$$|sinx| = \sqrt{1 = cos^2x}$$
or defined piecewisely = {$$\sin\left({x}\right)$$ in regions where it is above the x-axis, and -$$\sin\left({x}\right)$$ in regions where x is below the x-axis.
Is it possible to even truly isolate "$$\sin\left({x}\right)$$" from
$$sin^2x = 1 - cos^2x$$? It seems as the |$$\sin\left({x}\right)$$| is the closest we can to isolate it.
Sorry if I may seem confusing, but the concept of absolute value still confuses me.