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Chapter 1: Problem 34

Name the quadrant in which each angle lies. $$ -200^{\circ} $$

Short Answer

Expert verified
Quadrant II

Step by step solution

01

Understanding the Angle Measurement

Angles in standard position are measured counterclockwise from the positive x-axis. A full circle or one complete revolution is 360掳.
02

Determine the Equivalent Positive Angle

To understand where -200掳 lies, convert it into a positive angle by adding 360掳: -200掳 + 360掳 = 160掳.
03

Identify the Quadrant

Now that the angle is converted to 160掳, identify which quadrant it lies in by examining the unit circle. 0掳 to 90掳 lies in Quadrant I, 90掳 to 180掳 lies in Quadrant II. So 160掳 is in Quadrant II.

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Key Concepts

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

Angle Measurement
Angles are fundamental in trigonometry, and it is important to understand how they are measured. Angles in standard position are always measured starting from the positive x-axis. This is the horizontal axis on the coordinate plane. Imagine tracing around a circle. You start from the right and move counterclockwise. One complete circle is 360 degrees (掳). If you go clockwise, the angle is negative. For example, -200掳 is an angle measured 200掳 in the clockwise direction.
When converting negative angles to positive, you simply add 360掳. So, -200掳 becomes 160掳. In general, for any negative angle, adding 360掳 gets you the equivalent positive angle.
Standard Position of Angles
In trigonometry, the concept of the 'standard position' of an angle is essential. An angle is in standard position when its vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis. The angle is measured counterclockwise from this axis to the terminal side.
This helps in easily determining in which quadrant the terminal side of an angle lies. For instance, for an angle of 160掳, the angle is measured from the positive x-axis counterclockwise. Since 160掳 is between 90掳 and 180掳, it lies in Quadrant II.
Using this concept helps in various applications and solves trigonometric problems efficiently.
Unit Circle
The unit circle is a valuable tool in trigonometry. It is a circle with a radius of 1 unit, centered at the origin of the coordinate plane.
The circumference of the unit circle corresponds to angles:
  • 0掳 at (1, 0)
  • 90掳 at (0, 1)
  • 180掳 at (-1, 0)
  • 270掳 at (0, -1)

When understanding which quadrant an angle lies in, the unit circle is your best reference. For example, an angle of 160掳 on the unit circle will be between 90掳 and 180掳 which is in Quadrant II.
This visualization is not only important for identifying quadrants but also in understanding sine and cosine values and trigonometric identities.

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