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Chapter 1: Problem 8

State in which quadrant or on which axis the given angle lies. $$2009^{\circ}$$

Short Answer

Expert verified
The angle 2009掳 lies in Quadrant I.

Step by step solution

01

- Understand the Angle

An angle of 2009掳 is greater than 360掳, which means it makes multiple full turns around the circle. To determine where it lies, we need to find its equivalent angle between 0掳 and 360掳.
02

- Use Modulo Operation

Use the modulo operation to find the equivalent angle between 0掳 and 360掳. Calculate: 2009 mod 360.
03

- Calculate the Modulo

So, 2009掳 is equivalent to 49掳.
04

- Determine the Quadrant

Angles between 0掳 and 90掳 lie in Quadrant I. Since 49掳 is between 0掳 and 90掳, 2009掳 lies in Quadrant I.

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Key Concepts

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

Angle Reduction
When faced with angles larger than 360掳, we need to reduce them to a simpler form. This process is called angle reduction. Large angles, like 2009掳, make multiple full turns around a circle, each turn representing 360掳. By finding an equivalent angle within a single 360掳 rotation, we simplify our problems significantly.
For example:
  • If you have an angle of 720掳, it completes exactly 2 full turns (720掳 / 360掳 = 2) and ends at 0掳.
  • An angle of 450掳 is like taking a full turn (360掳) and then adding 90掳.
Reducing angles helps us determine their positions easily. We can then work within the familiar 0掳 to 360掳 range instead of dealing with enormous numbers.
Quadrants in Trigonometry
The coordinate system is divided into four quadrants. These quadrants help us understand direction and position of angles and lines.
Here's a quick breakdown:
  • Quadrant I: Angles between 0掳 and 90掳. Both x and y coordinates are positive here.
  • Quadrant II: Angles between 90掳 and 180掳. The x coordinate is negative, and the y coordinate is positive.
  • Quadrant III: Angles between 180掳 and 270掳. Both x and y coordinates are negative.
  • Quadrant IV: Angles between 270掳 and 360掳. The x coordinate is positive, and the y coordinate is negative.
For example, consider the angle 49掳. It is between 0掳 and 90掳, landing it in Quadrant I. This information helps us understand that 49掳 (and its equivalent angles, like 2009掳) lie in Quadrant I.
Modulo Operation
To simplify large angles, we use the modulo operation. This mathematical process finds the remainder when one number is divided by another, typically making our calculations easier.
For angles, we often want the remainder when the angle is divided by 360掳. This tells us the equivalent angle in a single circle rotation.
For example:
  • When we calculate 2009 mod 360, we are looking for the remainder of 2009 divided by 360.
  • The calculation gives us 2009 梅 360 = 5 R 49, where R represents the remainder.
So, the equivalent angle of 2009掳 in a single 360掳 rotation is 49掳. Using the modulo operation simplifies larger problems, allowing us to work within standard, more manageable angle ranges.

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