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Chapter 1: Problem 4

State in which quadrant or on which axis the given angle lies. $$-313^{\circ}$$

Short Answer

Expert verified
-313掳 lies in the first quadrant.

Step by step solution

01

- Understand negative angles

Negative angles are measured in the clockwise direction from the positive x-axis.
02

- Convert negative angle to positive

Add 360掳 to the negative angle until it is positive. For \(-313掳\), \(-313掳 + 360掳 = 47掳\).
03

- Determine the quadrant

Angles between \(0掳\) and \(90掳\) lie in the first quadrant. Since \(47掳\) is between \(0掳\) and \(90掳\), \(-313掳\) lies in the first quadrant.

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Key Concepts

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

negative angles
In trigonometry, angles can be positive or negative. While positive angles are measured counterclockwise from the positive x-axis, negative angles are measured clockwise. This is important to remember because the direction of measurement affects the final position of the angle on the coordinate plane.
Negative angles might seem tricky at first, but they follow the same principles as positive angles. To visualize a negative angle, simply start at the positive x-axis and rotate clockwise the specified number of degrees. This helps in understanding the angle's final position in the various quadrants.
angle conversion
Converting negative angles to positive ones can simplify determining their position on the coordinate plane. The simplest way is to keep adding 360掳 to the negative angle until you get a positive one.
For example, with \( -313掳 \), adding 360掳 once gives: \( -313掳 + 360掳 = 47掳 \). Since \( 47掳 \) is a positive angle, conversion is complete.
Converting angles ensures they fall within the 0掳 to 360掳 range, making it easier to find their respective quadrant. If the converted angle exceeds 360掳, you can keep subtracting 360掳 until it falls within the 0掳 to 360掳 range.
trigonometric quadrants
The coordinate plane is divided into four quadrants, each representing a specific range of angles:
  • First Quadrant: 0掳 to 90掳
  • Second Quadrant: 90掳 to 180掳
  • Third Quadrant: 180掳 to 270掳
  • Fourth Quadrant: 270掳 to 360掳

For \( 47掳 \), it falls within the first quadrant (0掳 to 90掳). Thus, the original angle \( -313掳 \) also lies in the first quadrant after conversion.
Understanding these quadrants helps in determining where an angle lies and identifies its trigonometric function values' signs. Each quadrant represents different signs (+/-) for sine, cosine, and tangent functions, which are essential for solving advanced trigonometry problems.

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