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I'm trying to find an angle between 0 and 2π that is coterminal with -4π/3 in terms of pi.

How do I go about doing this?

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Alright, I came up with 2/3π. Hopefully that's right. Thanks guys. – Brandt Oct 28 '12 at 22:35
up vote 2 down vote accepted

In general, you'll add or subtract $2\pi$ until it ends up in the desired range.

Rather than do this one at a time, if we want to find some angle coterminal with $\theta$ in the interval $[\theta_1,\theta_2]$, write $$\theta_1\leq\theta+2k\pi\leq\theta_2,$$ and find $k\in\Bbb Z$ so that this works. Once you've found this, $\theta+2k\pi$ will then be the desired angle.

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Hint: The angles $\theta$ and $\phi$ are coterminal iff they differ by an integer multiple of $2\pi$. So try to find a $k$ such that $-\dfrac{4\pi}{3}$ is between $0$ and $2\pi$. It will not take long.

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First, do you see that $-4 \pi/3$ is in Quadrant II? Drop a vertical line down from this terminal side to touch the negative $x$-axis. Since $-4 \pi/3$ is a little more than $-\pi$ you can use your sketch or think "$-\pi$ plus what gives $-4 \pi/3$ ? Keep in mind that reference angles are defined to be (positive) acute angles: $0<\alpha <0$. Quite often reference angles come out to $\pi/6, \pi/4$, or $\pi/3$.

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