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Chapter 1: Problem 6

State in which quadrant or on which axis the given angle lies. $$621^{\circ}$$

Short Answer

Expert verified
Third quadrant

Step by step solution

01

Understand the Full Circle

A full circle represents an angle of 360 degrees. The position of an angle in a circle can be determined by reducing it modulo 360.
02

Reduce the Angle

To find the equivalent angle between 0 and 360 degrees, reduce 621 modulo 360. Calculate the remainder of 621 divided by 360 using the formula: 621 mod 360.
03

Perform the Calculation

Divide 621 by 360:621 梅 360 = 1 remainder 261So, 621 degrees is equivalent to 261 degrees because 621 - 360 = 261.
04

Identify the Quadrant or Axis

An angle of 261 degrees lies in the third quadrant since it is between 180 degrees and 270 degrees.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

full circle
A full circle represents a rotation of 360 degrees. This concept is foundational in trigonometry because it helps us understand how angles wrap around the circle. When an angle exceeds 360 degrees, it means the angle has completed one or more full rotations. For instance, 720 degrees equals two full circles because 720 is twice 360. Understanding this helps us simplify and interpret larger angles by reducing them to a value between 0 and 360 degrees.
modulo operation
The modulo operation, often noted as 'mod', helps us find the remainder when one number is divided by another. In trigonometry, it is particularly useful for reducing angles. For example, to reduce 621 degrees to an equivalent angle in a standard circle, we calculate 621 mod 360. The operation tells us how many degrees past the nearest full circle the angle extends. In this case, 621 梅 360 equals 1 remainder 261, meaning that 621 degrees is equivalent to 261 degrees after accounting for one full rotation.
angle reduction
Angle reduction involves using the modulo operation to convert large angles to a standard range, typically between 0 and 360 degrees. This is done to simplify the understanding and analysis of the angle. For example, the angle 621 degrees can be reduced by subtracting multiples of 360:
  • 621 - 360 = 261 degrees.
This process helps us identify which quadrant or axis the angle is located in.
quadrants
The coordinate plane is divided into four quadrants, each representing a 90-degree section of the circle. The quadrants help us determine the sign and properties of trigonometric functions:
  • First Quadrant: 0 to 90 degrees.
  • Second Quadrant: 90 to 180 degrees.
  • Third Quadrant: 180 to 270 degrees.
  • Fourth Quadrant: 270 to 360 degrees.
An angle of 261 degrees falls in the third quadrant, lying between 180 and 270 degrees. Each quadrant has specific characteristics for trigonometric functions, which can affect their signs and values.

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