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Chapter 1: Problem 37

Name the quadrant in which each angle lies. $$ 933^{\circ} $$

Short Answer

Expert verified
\(933^{\text{掳}}\) lies in Quadrant III since the equivalent angle is \(213^{\text{掳}}\).

Step by step solution

01

Understand the problem

Determine the quadrant where the angle lies by first converting the large given angle to an effective angle within the standard range of 0 to 360 degrees. Quadtrants are key areas in the coordinate system, divided by the X-axis and Y-axis.
02

Find the Equivalent Angle

First, reduce the large angle. The angle provided is \(933^{\text{掳}}\). Angles are periodic every \(360^{\text{掳}}\), so subtract \(360^{\text{掳}}\) from \(933^{\text{掳}}\) repeatedly until the result is between \(0^{\text{掳}}\) and \(360^{\text{掳}}\). Solve: \[ 933^{\text{掳}} - 2 \times 360^{\text{掳}} = 933^{\text{掳}} - 720^{\text{掳}} = 213^{\text{掳}} \]
03

Determine the Quadrant

The equivalent angle is \(213^{\text{掳}}\). To identify the quadrant in which this angle lies, note that each quadrant represents: - Quadrant I: \(0^{\text{掳}} < \theta < 90^{\text{掳}}\)- Quadrant II: \(90^{\text{掳}} < \theta < 180^{\text{掳}}\)- Quadrant III: \(180^{\text{掳}} < \theta < 270^{\text{掳}}\)- Quadrant IV: \(270^{\text{掳}} < \theta < 360^{\text{掳}}\)Given that \(213^{\text{掳}}\) is between \(180^{\text{掳}}\) and \(270^{\text{掳}}\), \(213^{\text{掳}}\) falls in Quadrant III.

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Key Concepts

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

Quadrants
In the Cartesian coordinate system, the plane is divided into four quadrants. These quadrants help to determine the positions and properties of angles and points.
  • Quadrant I: This is the upper right section where both x and y values are positive (0掳 < 胃 < 90掳).
  • Quadrant II: Located in the upper left, x is negative, and y is positive (90掳 < 胃 < 180掳).
  • Quadrant III: The lower left section where both x and y are negative (180掳 < 胃 < 270掳).
  • Quadrant IV: This is the lower right, where x is positive, but y is negative (270掳 < 胃 < 360掳).

Each quadrant affects the signs of trigonometric functions. Angles like 933掳 can be large, but knowing in which quadrant the angle falls helps to better understand its properties and simplify computations. In this case, reducing 933掳 down to 213掳 shows it lies in Quadrant III.
Angle Reduction
Angle reduction is a critical skill in trigonometry, especially when dealing with angles outside the standard 0掳 to 360掳 range.
One can reduce any angle to this range using modulo operation, which finds the remainder after division by 360掳.

Following steps can simplify this:
  • Subtract 360掳 repetitively: For example, given 933掳, subtract 360掳:
    933掳 - 720掳 = 213掳.
  • Using modulo operation: The same result can be achieved more straightforwardly with modulo: ewline 933掳 % 360掳 = 213掳.


Effective angle reduction simplifies identifying the corresponding quadrant and further calculations for trigonometric functions.
Trigonometric Functions
Understanding trigonometric functions within different quadrants is key. Here鈥檚 a brief overview of how sine, cosine, and tangent function values change:
  • Quadrant I: All functions (sine, cosine, tangent) are positive.
  • Quadrant II: Sine is positive, but cosine and tangent are negative.
  • Quadrant III: Tangent is positive, whereas sine and cosine are negative.
  • Quadrant IV: Cosine is positive, with sine and tangent being negative.

When dealing with large angles like 933掳, reducing it first helps determine which quadrant the angle falls into, and thus, compute function values accurately. For the reduced angle 213掳 in Quadrant III, sine and cosine would be negative, while the tangent function remains positive.

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