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Chapter 1: Problem 31

Name the quadrant in which each angle lies. $$ 85^{\circ} $$

Short Answer

Expert verified
85掳 is in Quadrant I.

Step by step solution

01

Understand the Quadrants

Divide the coordinate plane into four quadrants. Quadrant I is from 0掳 to 90掳, Quadrant II is from 90掳 to 180掳, Quadrant III is from 180掳 to 270掳, and Quadrant IV is from 270掳 to 360掳.
02

Identify the Range of the Angle

The given angle is 85掳. Determine which quadrant 85掳 falls into by checking the quadrant ranges. Compare 85掳 with the upper and lower bounds of each quadrant.
03

Determine the Quadrant

85掳 lies between 0掳 and 90掳, so it falls in Quadrant I.

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Key Concepts

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

Coordinate Plane Quadrants
The coordinate plane is divided into four sections called quadrants. Each quadrant has its own unique range of angles. This makes it easier to identify where an angle lies.
The quadrants are labeled I, II, III, and IV in a counterclockwise direction, starting from the top-right section.
Here鈥檚 a breakdown of each quadrant:
  • 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

Understanding these ranges helps with quickly identifying the position of angles on the coordinate plane.
Angle Measurement
Measuring angles in degrees is fundamental to understanding their position on the coordinate plane.
Angles are measured starting from the positive x-axis and moving counterclockwise. This method ensures consistency and avoids confusion.
For instance, an angle of 85掳 starts from the positive x-axis and moves 85掳 counterclockwise. Always keep the direction in mind as it helps determine which quadrant an angle belongs to.
When given an angle, always compare it against the quadrant ranges to easily identify its placement.
This can be visualized as follows: if an angle lies between 0掳 and 90掳, it is in Quadrant I, and so on for the other quadrants.
Trigonometry Basics
Trigonometry deals with the relationships between angles and sides of triangles.
A fundamental part of understanding trigonometry is knowing the position of angles on the coordinate plane.
Identifying the correct quadrant of an angle is essential for solving trigonometric problems accurately.
  • Each trigonometric function (sine, cosine, tangent, etc.) behaves differently depending on the quadrant.
  • For example, cosine is positive in Quadrants I and IV, while sine is positive in Quadrants I and II.

Using the quadrant system helps in remembering these properties and applying them correctly in calculations.
Always begin your trigonometry analysis by determining in which quadrant the angle lies, as this will guide the rest of your problem-solving process.

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Most popular questions from this chapter

Find the degree measure of the smallest positive angle that is coterminal with each angle. $$ 540^{\circ} $$

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Solve each problem. Find \(\sin (\alpha),\) given that \(\cos (\alpha)=2 / 5\) and \(\sin (\alpha)<0\).

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A(n) __________ is one sixtieth of a minute.
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