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Chapter 5: Problem 10

Sketch the angles in standard position. $$210^{\circ}$$

Short Answer

Expert verified
Angle \(210^\circ\) in standard position will fall in the third quadrant when sketched in the x-y plane.

Step by step solution

01

Understanding the Angle Measurement

Firstly, recognize that \(210^\circ\) falls between \(180^\circ\) and \(270^\circ\). This indicates that the angle will lie in the third quadrant.
02

Draw a Horizontal Line

Create a graph with a horizontal (x-axis) and vertical line (y-axis). Then draw a horizontal straight line to the right, which represents \(0^\circ\) or \(360^\circ\).
03

Draw The Angle

From the horizontal line, measure and draw an angle of \(210^\circ\) starting counterclockwise from the positive x-axis until you reach the third quadrant. The terminal side of this angle lands in the third quadrant. The ray forming this angle from the origin makes a \(210^\circ\) angle with the positive x-axis.

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Key Concepts

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

Quadrants in a Coordinate System
The coordinate system is divided into four quadrants, each identified by the positive or negative direction of the x and y coordinates. These four quadrants are crucial in understanding the positioning and direction of angles in a standard position.
  • First Quadrant: Both x and y are positive. It ranges from 0 to 90 degrees.
  • Second Quadrant: The x is negative, y is positive. It covers from 90 to 180 degrees.
  • Third Quadrant: Both x and y are negative. It spans from 180 to 270 degrees.
  • Fourth Quadrant: The x is positive, y is negative. This includes angles from 270 to 360 degrees.
Knowing which quadrant an angle is in helps you to determine the signs of sine, cosine, and tangent functions. For example, an angle of 210 degrees, as in the exercise, falls in the third quadrant, indicating that both its sine and cosine values are negative.
Angle Measurement
When dealing with angles, understanding how to measure them is key. Angles in standard position start with their initial side along the positive x-axis.
  • Direction of Measurement: Angles are measured counterclockwise from the positive x-axis to the terminal side.
  • Full Circle: A complete rotation around the circle is 360 degrees.
  • Half Circle: A half circle, measuring 180 degrees, signifies a straight line from the right to the left along the x-axis.
In the exercise, the 210-degree angle begins at the positive x-axis and is measured counterclockwise. This specific measurement leads the angle to end in the third quadrant after surpassing 180 degrees.
Graphing Angles
Graphing angles involves visualizing an angle on the coordinate plane. This is crucial for analyzing the trigonometric functions associated with angles.
  • Initial Setup: Start by drawing the x and y axes, which establish your coordinate plane.
  • Placing the Angle: Draw a horizontal line to the right representing 0 degrees.
  • Angle Positioning: From the 0-degree line, measure counterclockwise to reach the desired angle.
For a 210-degree angle, begin from the positive x-axis, move counterclockwise past the 180-degree mark, stopping in the third quadrant. The endpoint, called the terminal side, shows where the side of the angle falls on the plane. Visualizing angles in this way simplifies interpreting trigonometric values like sine or cosine based on their positioning.

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Most popular questions from this chapter

This set of exercises will draw on the ideas presented in this section and your general math background. Find all values of \(t\) in \([0,2 \pi)\) such that \(\sin t=\cos t\)

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