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Chapter 5: Problem 27

Name the quadrant in which each angle lies. $$750^{\circ}$$

Short Answer

Expert verified
750掳 lies in Quadrant I.

Step by step solution

01

Identify the Quadrant Cycle

First, recognize that each complete cycle of angles is 360 degrees. Angles that exceed 360 degrees have completed one or more full revolutions.
02

Determine Equivalent Angle

To find the equivalent angle within the initial 360-degree cycle, divide the given angle by 360 degrees and find the remainder. Compute: \[ 750^{\text{掳}} \text{ mod } 360^{\text{掳}} \]. This means: \[ 750 - 2 \times 360 = 750 - 720 = 30 \text{掳} \]. So, \( 750^{\text{掳}} \) is equivalent to \( 30^{\text{掳}} \).
03

Determine the Quadrant

Knowing that \( 30^{\text{掳}} \) is a positive acute angle, it lies in the first quadrant (I). In trigonometric terms, the first quadrant ranges from \( 0^{\text{掳}} \) to \( 90^{\text{掳}} \).

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Key Concepts

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

Angle Cycles
Understanding angle cycles is key in trigonometry. An angle cycle refers to a complete rotation around a circle, which is 360 degrees. When angles exceed 360 degrees, they have completed one or more full revolutions. This is similar to wrapping around a circle multiple times. For instance, if an angle is 750 degrees, it means it has gone around the circle twice (720 degrees) and then an additional 30 degrees. This helps in simplifying and understanding large angles in terms of a smaller, more familiar range.
Equivalent Angles
Equivalent angles are angles that are in the same position within the cycle of 0 to 360 degrees. After we find how many full cycles an angle has completed, we can determine the equivalent angle within the initial cycle. For example, with a 750-degree angle, we calculate how many full cycles it has completed:
  • 750 梅 360 = 2 remainder 30
Thus, 750 degrees can be reduced to its equivalent angle of 30 degrees. This process makes it easier to analyze and work with angles, as the equivalent angle remains in the initial cycle of 0-360 degrees.
Quadrant Determination
Determining the quadrant of an angle helps to understand its position on the Cartesian plane. The Cartesian plane is divided into four quadrants:
  • First Quadrant: 0 to 90 degrees
  • Second Quadrant: 90 to 180 degrees
  • Third Quadrant: 180 to 270 degrees
  • Fourth Quadrant: 270 to 360 degrees
To determine the quadrant of an angle, we first find its equivalent angle. For instance, 750 degrees is equivalent to 30 degrees. Since 30 degrees lies between 0 to 90 degrees, it is in the first quadrant. This helps in identifying the position and behavior of the angle in trigonometric applications.

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