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Chapter 1: Problem 32

Name the quadrant in which each angle lies. $$ 110^{\circ} $$

Short Answer

Expert verified
Quadrant II

Step by step solution

01

Understand the quadrants

In the Cartesian plane, there are four quadrants: - Quadrant I: Angles between 0掳 and 90掳 - Quadrant II: Angles between 90掳 and 180掳 - Quadrant III: Angles between 180掳 and 270掳 - Quadrant IV: Angles between 270掳 and 360掳
02

Identify the given angle

The given angle is \( 110^\circ \).
03

Locate the angle in the appropriate quadrant

To determine in which quadrant \( 110^\circ \) lies, compare it to the ranges of the quadrants: \( 110^\circ \) is between \( 90^\circ \) and \( 180^\circ \), so it lies 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
Understanding how to measure angles is vital in trigonometry. Angles are measured in degrees, and a full circle is 360 degrees. Each angle measurement corresponds to a position on the Cartesian plane. We typically measure angles starting from the positive x-axis and moving counterclockwise.

When we say an angle is 110 degrees, we mean it rotates 110 degrees from the positive x-axis in a counterclockwise direction. Here's a quick guide:
  • 0掳 is on the positive x-axis.
  • 90掳 is on the positive y-axis.
  • 180掳 is on the negative x-axis.
  • 270掳 is on the negative y-axis.

    • Understanding this helps us determine in which quadrant an angle lies.
Cartesian Plane
The Cartesian plane is a two-dimensional plane defined by an x-axis and a y-axis. It is divided into four quadrants. Each quadrant helps us understand the location of angles and points:
  • Quadrant I: Located in the top-right section, where both x and y coordinates are positive.
  • Quadrant II: In the top-left, x is negative, and y is positive.
  • Quadrant III: Located in the bottom-left, where both x and y are negative.
  • Quadrant IV: In the bottom-right, x is positive, and y is negative.

    • Knowing these quadrants is essential in trigonometry and geometry. For example, an angle of 110 degrees is in Quadrant II because it falls between 90掳 (positive y-axis) and 180掳 (negative x-axis).
Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It involves functions such as sine, cosine, and tangent, which are used to relate the angles of a triangle to the lengths of its sides. Understanding the fundamentals, like angle measurement and the Cartesian plane, gives a solid foundation for more advanced topics in trigonometry.

For instance, knowing which quadrant an angle lies in helps determine the signs (+ or -) of its sine, cosine, and tangent values. In our case, an angle of 110 degrees is in Quadrant II, where sine is positive, and both cosine and tangent are negative. This understanding is crucial for solving trigonometric problems efficiently.

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