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Chapter 1: Problem 21

Find the reference angle for the given angle. $$275^{\circ}$$

Short Answer

Expert verified
The reference angle is $$85^{\text{掳}}.

Step by step solution

01

- Determine the Quadrant

First, identify which quadrant the given angle $$275^{\text{掳}}$$ is in. Since it is between $$270^{\text{掳}} \text{and} 360^{\text{掳}}, it lies in the fourth quadrant.
02

- Calculate Reference Angle

When the angle lies in the fourth quadrant, the reference angle can be found by subtracting the given angle from $$360^{\text{掳}}$$. $$\text{Reference Angle} = 360^{\text{掳}} - 275^{\text{掳}} $$
03

- Perform the Subtraction

Subtract $$275^{\text{掳}}$$ from $$360^{\text{掳}}$$ $$360^{\text{掳}} - 275^{\text{掳}} = 85^{\text{掳}}$$.

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Key Concepts

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

Reference Angle
A reference angle is the smallest angle between the terminal side of the given angle and the x-axis. It's always positive and is between 0掳 and 90掳. Finding the reference angle helps in simplifying trigonometric calculations.
To determine a reference angle, you need to know in which quadrant the original angle lies. This helps in understanding whether you subtract the angle from 180掳, 360掳, or any other specific values. For instance, for an angle of 275掳, it falls in the fourth quadrant. So, the reference angle is calculated using specific steps.
Quadrants
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

In this problem, the given angle is 275掳. Since 275掳 lies between 270掳 and 360掳, it is in the fourth quadrant. Knowing which quadrant an angle is in helps determine how to calculate the reference angle. Quadrants also indicate the sign (positive or negative) for the values of sine, cosine, and tangent.
Angle Subtraction
Subtracting angles is often necessary to find reference angles, especially when dealing with quadrants beyond the first. In the fourth quadrant, to find a reference angle, you subtract the given angle from 360掳. Here's the specific calculation step-by-step:
1. Identify the quadrant: 275掳 is in the fourth quadrant.
2. Use the reference angle formula: Reference Angle = 360掳 - Given Angle.
3. Perform the subtraction: 360掳 - 275掳 = 85掳.
This makes the reference angle for 275掳 equal to 85掳.
Understanding angle subtraction ensures you can find the correct reference angle for any given problem.

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Most popular questions from this chapter

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