Identify the quadrant that the given angle is located in. π 7 \frac{\pi}{7} 7 π radians

In this problem we're asked to identify what quadrant the given angle is in and here we're given the angle in radians pi over seven radians. Now we're gonna use two things here, our knowledge of fractions and angles that we already know on our unit circle, so let's go ahead and get started. Now looking at this angle pi over seven, if I consider all of the angles that I know on my unit circle, I know that I start at zero and then I rotate around to my first quadrantal angle, the one that is directly in between quadrant one and quadrant two, and this angle is pi over two. Now whenever I have a fraction, if my denominator is larger than some other fraction and they have the same exact numerator, I know that that angle has to be smaller. So this pi over two, since this has a denominator of two and my given angle has a denominator of seven and they both have that numerator of pi, I know that pi over seven is less than pi over two. So that tells me that my angle sits in between zero and pi over two on my unit circle. So it is right here in quadrant one. So pi over seven radians is an angle that is in quadrant one of my unit circle. Thanks for watching, and let me know if you have questions.

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

Business Calculus

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

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Multiple Choice

Identify the quadrant that the given angle is located in.
$\frac{\pi}{7}$ radians

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

Verified step by step guidance

Convert the given angle $ \frac{\pi}{7} $ radians into degrees for easier interpretation, if necessary. Use the conversion formula $ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} $.

Understand that a full circle is $ 2\pi $ radians (or 360 degrees), and each quadrant represents a quarter of the circle: Quadrant I (0 to $ \frac{\pi}{2} $), Quadrant II ($ \frac{\pi}{2} $ to $ \pi $), Quadrant III ($ \pi $ to $ \frac{3\pi}{2} $), and Quadrant IV ($ \frac{3\pi}{2} $ to $ 2\pi $).

Compare $ \frac{\pi}{7} $ to the boundaries of the quadrants. Since $ \frac{\pi}{7} $ is a small positive angle, it lies between 0 and $ \frac{\pi}{2} $, which corresponds to Quadrant I.

Verify your conclusion by considering the unit circle. In Quadrant I, both the sine and cosine of the angle are positive, which aligns with the properties of $ \frac{\pi}{7} $.

Conclude that the angle $ \frac{\pi}{7} $ radians is located in Quadrant I.