To rewrite the expression [tex]\( \sqrt{3} \sin(x) - \cos(x) \)[/tex] in terms of sine only, we can use the Pythagorean identity, which relates [tex]\(\cos(x)\)[/tex] and [tex]\(\sin(x)\)[/tex].
1. Recall the Pythagorean identity:
[tex]\[
\cos^2(x) + \sin^2(x) = 1
\][/tex]
2. Solve for [tex]\(\cos(x)\)[/tex]:
[tex]\[
\cos(x) = \sqrt{1 - \sin^2(x)}
\][/tex]
Note that [tex]\(\cos(x)\)[/tex] can also be [tex]\(-\sqrt{1 - \sin^2(x)}\)[/tex]. However, for the purpose of rewriting the given expression, we will initially use the positive root.
3. Substitute [tex]\(\cos(x)\)[/tex] in the original expression:
[tex]\[
\sqrt{3} \sin(x) - \cos(x) \implies \sqrt{3} \sin(x) - \sqrt{1 - \sin^2(x)}
\][/tex]
4. Simplify further if possible:
This is the intermediate form:
[tex]\[
\sqrt{3} \sin(x) - \sqrt{1 - \sin^2(x)}
\][/tex]
However, we need to consider both possible signs for cosine to ensure proper representation. Hence, we should also consider:
[tex]\[
\sqrt{3} \sin(x) - (-\sqrt{1 - \sin^2(x)}) = \sqrt{3} \sin(x) + \sqrt(1 - \sin^2(x))
\][/tex]
5. Substitute back to ensure the simplified version:
Given the form:
[tex]\[
\sqrt{3} \sin(x) - \sqrt(1 - \sin^2(x))
\][/tex]
This indeed simplifies as:
[tex]\[
-\sqrt{\cos^2(x)} + \sqrt{3}\sin(x)
\][/tex]
Thus, the expression rewritten in terms of sine only is:
[tex]\[
\sqrt{3} \sin(x) - \sqrt{1 - \sin^2(x)}
\][/tex]
And its simplified version is:
[tex]\[
-\sqrt{\cos(x)^2} + \sqrt{3} \sin(x)
\][/tex]
