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Chapter 7: Problem 44

Find the reference angle of each angle. $$ 210^{\circ} $$

Short Answer

Expert verified
The reference angle is 30掳.

Step by step solution

01

Identify the Quadrant

Determine in which quadrant the given angle lies. Since 210掳 is between 180掳 and 270掳, it is in the third quadrant.
02

Use Reference Angle Formula for the Third Quadrant

In the third quadrant, the reference angle is found by subtracting 180掳 from the given angle. So, compute 210掳 - 180掳.
03

Perform the Calculation

Calculate the reference angle: 210掳 - 180掳 = 30掳.

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Key Concepts

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

Reference Angles
A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis.
Reference angles are always between 0掳 and 90掳, inclusive. They help simplify the calculation of trigonometric functions for any angle.
The reference angle can be found for any given angle, whether it is positive or negative, by following specific rules depending on the quadrant in which the angle is located.
For instance, if we have an angle in the fourth quadrant, its reference angle would be calculated by subtracting it from 360掳. If the angle was, say, 300掳, the reference angle would be: 360掳 - 300掳 = 60掳. Reference angles are great tools because they allow us to use familiar trigonometric values even for large or negative angles.
Trigonometric Quadrants
The coordinate plane is divided into four quadrants. Understanding which quadrant an angle lies in helps in calculating its reference angle.
Here's a quick rundown of the quadrants and the general range of angles in each:
  • First Quadrant: 0掳 to 90掳 (angles are positive)
  • Second Quadrant: 90掳 to 180掳 (angles are positive)
  • Third Quadrant: 180掳 to 270掳 (angles are positive)
  • Fourth Quadrant: 270掳 to 360掳 (angles are positive)

For example, an angle of 210掳 is in the third quadrant because it falls between 180掳 and 270掳. Knowing this helps us quickly determine which formula to use for finding the reference angle.
Angle Subtraction
Once you know the quadrant an angle falls in, you can then use angle subtraction to find the reference angle.
Each quadrant has its own formula for this:
  • First Quadrant: The reference angle is the angle itself.
  • Second Quadrant: The reference angle is calculated as 180掳 - angle.
  • Third Quadrant: The reference angle is calculated as angle - 180掳.
  • Fourth Quadrant: The reference angle is calculated as 360掳 - angle.

In the earlier example, 210掳 is in the third quadrant. The reference angle is found by subtracting 180掳 from 210掳:
\[ 210^\text{掳} - 180^\text{掳} = 30^\text{掳} \]
This angle (30掳) is now easy to work with in terms of sine, cosine, and tangent functions.

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