Identify the reference angle of each given angle. 7 π 4 \frac{7\pi}{4} 4 7 π rad

In this problem, we're asked to identify the reference angle of the given angle, and the angle that we're given here is 7 pi over 4. Now you could choose just to kind of memorize what the reference angle is here by looking at that denominator because I know that the denominator of its reference angle is the exact same 4 up here in quadrant 1, pi over 4. But you can also think about this by actually going to take a look at your angle on the unit circle and measuring it here from the nearest point of the x axis. Now from that nearest point of the x axis, I know that it is pi over 4 radians or 45 degrees away from that nearest point on the x axis. So that tells me that the reference angle of 7 pi over 4 is pi over 4 radians or 45 degrees. Thanks for watching and let me know if you have questions.

3. Unit Circle

Reference Angles

Trigonometry

3. Unit Circle

Reference Angles

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Multiple Choice

Identify the reference angle of each given angle.
$\frac{7\pi}{4}$ rad

$\frac{\pi}{6}$

$\frac{\pi}{4}$

$\frac{\pi}{3}$

Verified step by step guidance

Step 1: Understand that the reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. It is always positive and less than or equal to $ \frac{\pi}{2} $ radians.

Step 2: For the angle $ \frac{7\pi}{4} $, first determine its position in the unit circle. Since $ \frac{7\pi}{4} $ is greater than $ 2\pi $, subtract $ 2\pi $ to find its equivalent angle within the first full rotation: $ \frac{7\pi}{4} - 2\pi = \frac{7\pi}{4} - \frac{8\pi}{4} = -\frac{\pi}{4} $. The reference angle is $ \frac{\pi}{4} $.

Step 3: For the angle $ \frac{\pi}{6} $, since it is already between 0 and $ \frac{\pi}{2} $, it is its own reference angle. Therefore, the reference angle is $ \frac{\pi}{6} $.

Step 4: For the angle $ \frac{\pi}{3} $, similarly, since it is also between 0 and $ \frac{\pi}{2} $, it is its own reference angle. Therefore, the reference angle is $ \frac{\pi}{3} $.

Step 5: Summarize the reference angles: $ \frac{7\pi}{4} $ has a reference angle of $ \frac{\pi}{4} $, $ \frac{\pi}{6} $ has a reference angle of $ \frac{\pi}{6} $, and $ \frac{\pi}{3} $ has a reference angle of $ \frac{\pi}{3} $.