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Chapter 6: Problem 63

Find the reference angle and the exact function value if they exist. $$\csc 225^{\circ}$$

Short Answer

Expert verified
225^{\bullet} \text { is } -225-\text{ degrees}

Step by step solution

01

- Identify the given angle

The given angle is 225 degrees.
02

- Determine the quadrant

An angle of 225 degrees lies in the third quadrant because it is between 180 and 270 degrees.
03

- Calculate the reference angle

The reference angle for an angle in the third quadrant is found by subtracting 180 degrees from the angle:

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Key Concepts

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

Reference Angle
The reference angle is a fundamental concept in trigonometry. It represents the smallest angle formed by the terminal side of the given angle and the x-axis. For any given angle, the reference angle always lies between 0掳 and 90掳. This simplification helps us calculate the trigonometric functions more easily.

Let's calculate the reference angle for an angle that lies in different quadrants:
  • In the first quadrant (0掳 to 90掳), the reference angle is the angle itself.
  • In the second quadrant (90掳 to 180掳), subtract the angle from 180掳.
  • In the third quadrant (180掳 to 270掳), subtract 180掳 from the angle.
  • In the fourth quadrant (270掳 to 360掳), subtract the angle from 360掳.

Given the angle of 225掳, which falls in the third quadrant, we find the reference angle using the formula for the third quadrant:

\[\text{reference angle} = 225^\text{掳} - 180^\text{掳} = 45^\text{掳} \] Hence, the reference angle here is 45掳.
Third Quadrant
Angles in trigonometry can be located in four quadrants, each offering specifics about the sign and behavior of trigonometric functions.

When dealing with the third quadrant (between 180掳 and 270掳), it is important to remember:
  • Both sine (\(\text{sin}\)) and cosine (\(\text{cos}\)) values are negative because the coordinates on the unit circle are both negative in this quadrant.
  • The tangent (\(\text{tan}\)) function is positive because it is defined as the sine divided by the cosine, and a negative divided by a negative gives a positive result.

Given that 225掳 is located in this quadrant, we know:
\(\text{cos}225^\text{掳} = -\frac{\text{鈭�2}}{2}\) and \(\text{sin}225^\text{掳} = -\frac{\text{鈭�2}}{2}\) with positive \(\text{tan}225^\text{掳} = 1\). Since cosecant is the reciprocal of sine, we can proceed to find:
Cosecant
The cosecant function, written as \(\text{csc}\), is the reciprocal of the sine function. This relationship gives us an easy way to derive the value of the cosecant if we know the sine.

The formula for cosecant is:
\[\text{csc} \theta = \frac{1}{\text{sin} \theta} \] Given our problem, where \(\text{sin}225^\text{掳} = -\frac{\text{鈭�2}}{2}\), we find:
\[\text{csc}225^\text{掳} = \frac{1}{-\frac{\text{鈭�2}}{2}} = -\frac{2}{\text{鈭�2}} = -\text{鈭�2} \] Therefore, \(\text{csc}225^\text{掳}\) equals \(-\text{鈭�2}\). Understanding these relationships and calculations will help you grasp trigonometric functions involving the cosecant better.

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