Question

See question in screenshot below:

Answer 1

Below

Explanation:

Use trig identity:

sin (x-y) = sinx cosy - cosx siny

= sinx cos (4pi/3) - cos x sin (4pi/3)

= - 1/2 sin x + sqrt 3/2 * cos x

Answer 2

`-(1)/(2)\bigg(\sin\left(\text{x}\right)-√(3)\cos\left(\text{x}\right)\bigg)`

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Work Shown:

`\sin(A-B) = \sin(A)\cos(B)-\cos(A)\sin(B)\\\\\\\sin\left(\text{x}-(4\pi)/(3)\right) = \sin\left(\text{x}\right)\cos\left((4\pi)/(3)\right)-\cos\left(\text{x}\right)\sin\left((4\pi)/(3)\right)\\\\\\\sin\left(\text{x}-(4\pi)/(3)\right) = \sin\left(\text{x}\right)\left(-(1)/(2)\right)-\cos\left(\text{x}\right)\left(-(√(3))/(2)\right)\\\\\\`

`\sin\left(\text{x}-(4\pi)/(3)\right) = -(1)/(2)\sin\left(\text{x}\right)+(√(3))/(2)\cos\left(\text{x}\right)\\\\\\\sin\left(\text{x}-(4\pi)/(3)\right) = -(1)/(2)\bigg(\sin\left(\text{x}\right)-√(3)\cos\left(\text{x}\right)\bigg)\\\\\\`

You can a TI83 or TI84 calculator to confirm the answer. Or you can use graphing software tools like Desmos or GeoGebra.

Another approach to verification is using a table of values, but I don't really recommend this option.

The first line I mentioned is one of the many trig identities to either memorize or have on a reference sheet. I prefer the reference sheet option since there are many trig identities to keep track of.

Use the unit circle to determine that
`\cos\left((4\pi)/(3)\right) = -(1)/(2)` and
`\sin\left((4\pi)/(3)\right) = -(√(3))/(2)`. Look in quadrant III (which is the southwest quadrant).