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I need help finding the reference for these given angles...

(a) 120° (b) -210° (c) 780°

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do you need help visualizing the above angles? –  jay-sun Oct 29 '12 at 21:14
    
No...I need help finding the reference angles... –  Brandt Oct 29 '12 at 21:25
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According to a definition found through Google, "The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis" where the given angle is represented with its first ray along the positive $x$-axis. –  alex.jordan Oct 29 '12 at 21:28
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I would now draw each angle on a polar graph and apply the definition that alex.jordan gave (or whatever one is in your book if different). –  Ross Millikan Oct 29 '12 at 21:51
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1 Answer 1

The reference angle can be defined in various ways. One way is as follows: the reference angle is the acute angle formed by the terminal side of the given angle, and the $x$-axis, which was mentioned in the comments. You can play around with this visually on the coordinate plane to get a nice generalization:

Suppose $\theta$ is an angle, from $0^{\circ}$ to $360^{\circ}$.

$\bullet$ If $\theta$ is in the first quadrant, then the reference is itself, $\theta$.

$\bullet$ If $\theta$ is in the second quadrant, the reference is $180^{\circ} - \theta$.

$\bullet$ If $\theta$ is in the third quadrant, the reference is $\theta - 180^{\circ}$.

$\bullet$ If $\theta$ is in the fourth qudrant, the reference is $360^{\circ} - \theta$.

Now, $120^{\circ}$ is in the second quadrant, so it's reference angle would be $\ldots$

For your other two angles, they are less than $0^{\circ}$ or greater than $360^{\circ}$. Thus, you have to find the angle from $0^{\circ}$ to $360^{\circ}$ that coincides with each of them. To do this, you add or subtract $360^{\circ}$ until you get an angle that is between $0^{\circ}$ to $360^{\circ}$.

$-210^{\circ} + 360^{\circ} = 150^{\circ}$, and since $150^{\circ}$ is in the second quadrant, the reference of $-210^{\circ}$ is $\ldots$

$780^{\circ} - 360^{\circ} - 360^{\circ} = 60^{\circ}$, and since $60^{\circ}$ is in the first quadrant, the reference of $780^{\circ}$ is $\ldots$

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