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Chapter 1: Problem 9

State in which quadrant or on which axis the given angle lies. $$1079^{\circ}$$

Short Answer

Expert verified
Quadrant IV

Step by step solution

01

- Understand the Angles

Recall that the coordinate plane is divided into four quadrants. Quadrant I: 0掳 to 90掳, Quadrant II: 90掳 to 180掳, Quadrant III: 180掳 to 270掳, Quadrant IV: 270掳 to 360掳. Angles greater than 360掳 repeat in the same order as the described quadrants.
02

- Find the Equivalence of the Angle within 0掳 to 360掳

To find an equivalent angle within the range of 0掳 to 360掳, take the given angle 1079掳 and subtract multiples of 360掳: \[1079 \, ^{\circ} \mod 360 \, ^{\circ} = 1079 \, ^{\circ} - 3 \cdot 360 \, ^{\circ} = 1079 \, ^{\circ} - 1080 \, ^{\circ} = -1 \, ^{\circ}\]. Since angles are typically given in positive measure, -1掳 is equivalent to 359掳 (360掳 - 1掳).
03

- Determine the Quadrant

Since 359掳 lies in the interval 270掳 to 360掳, it is in Quadrant IV.

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Key Concepts

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

Angle Measurement
Understanding how angles are measured is fundamental to elementary trigonometry.
Angles are often measured in degrees (掳), where a full circle is 360掳.
Here's a quick recap:
  • 0掳 corresponds to the rightwards horizontal direction.
  • 90掳 points straight upwards.
  • 180掳 is directly opposite of 0掳, pointing left.
  • 270掳 points straight downwards.
Angles can be positive or negative:
  • Positive angles are measured counterclockwise from the positive x-axis.
  • Negative angles are measured clockwise from the positive x-axis.
This foundational understanding is crucial for solving problems related to angles and their positions in the coordinate plane.
Quadrants of the Coordinate Plane
The coordinate plane is divided into four quadrants:
  • Quadrant I: 0掳 to 90掳
  • Quadrant II: 90掳 to 180掳
  • Quadrant III: 180掳 to 270掳
  • Quadrant IV: 270掳 to 360掳
These quadrants help in understanding where an angle lies, based on its measurement. For example:
  • 45掳 lies in Quadrant I
  • 135掳 lies in Quadrant II
  • 225掳 lies in Quadrant III
  • 315掳 lies in Quadrant IV
This information is used to determine the position of given angles, even when their measures exceed 360掳 or are negative.
Knowing the respective ranges of these quadrants is instrumental when dealing with angles of any size.
Angle Equivalence Modulo 360 Degrees
Angles greater than 360掳 or less than 0掳 can be converted into an equivalent angle within the range of 0掳 to 360掳.
This concept is called 'angle equivalence modulo 360 degrees'.
In our example, we started with 1079掳:
  • First, we reduce 1079掳 to an angle less than 360掳 by subtracting multiples of 360掳.
  • This is done with the calculation: \[1079 \, ^{\circ} \mod 360 \, ^{\circ} = 1079 \, ^{\circ} - 3 \cdot 360 \, ^{\circ} = 1079 \, ^{\circ} - 1080 \, ^{\circ} = -1 \, ^{\circ}\]
  • Here, we see that -1掳 is the equivalent of 359掳, as \[-1掳 + 360掳 = 359掳\]
This angle (359掳) lies within Quadrant IV, as we determined previously. Using this method allows us to deal with any angle, regardless of its size.

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Most popular questions from this chapter

Find the values of the other five trigonometric functions of the acute angle \(A\) given the indicated value of one of the functions. $$\tan A=\frac{5}{9}$$

Find the exact values of \(\sin \theta\) and \(\cos \theta\) when \(\tan \theta\) has the indicated value. $$\tan \theta=0$$

Find the values of the other five trigonometric functions of the acute angle \(A\) given the indicated value of one of the functions. $$\sin A=\frac{3}{4}$$

Find the exact values of \(\cos \theta\) and \(\tan \theta\) when \(\sin \theta\) has the indicated value. $$\sin \theta=\frac{1}{2}$$

For an acute angle \(A\), can \(\cos A\) be larger than 1 ? Explain your answer.
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