Question
Mrs. Owen is trying to find cos 300° . She has opened the Unit Circle, but forgot how to use it. Explain to Mrs. Owen how to find cos 300° using the Unit Circle. Also include the answer in your response.
Answers
GPT-4o mini
To help Mrs. Owen find \( \cos 300^\circ \) using the Unit Circle, let's follow these steps:
1. **Understanding the Unit Circle:** The Unit Circle is a circle with a radius of 1 centered at the origin (0, 0) of a coordinate plane. The angle is measured from the positive x-axis, rotating counterclockwise for positive angles and clockwise for negative angles.
2. **Identifying the Angle:** The angle \( 300^\circ \) is greater than \( 270^\circ \) and less than \( 360^\circ \). This means it is located in the fourth quadrant of the Unit Circle.
3. **Finding the Reference Angle:** The reference angle is the acute angle formed with the x-axis. To find the reference angle for \( 300^\circ \), we can subtract it from \( 360^\circ \):
\[
360^\circ - 300^\circ = 60^\circ.
\]
Thus, the reference angle is \( 60^\circ \).
4. **Finding the Cosine Value:** In the Unit Circle, the coordinates of the angle \( 300^\circ \) correspond to \( (x, y) \), where \( x = \cos 300^\circ \) and \( y = \sin 300^\circ \). Since \( 300^\circ \) is in the fourth quadrant, the cosine value is positive.
5. **Using the Reference Angle:** The cosine of \( 300^\circ \) is the same as the cosine of its reference angle \( 60^\circ \):
\[
\cos 60^\circ = \frac{1}{2}.
\]
Since \( 300^\circ \) is in the fourth quadrant, we have:
\[
\cos 300^\circ = \cos 60^\circ = \frac{1}{2}.
\]
So, **the answer is**:
\[
\cos 300^\circ = \frac{1}{2}.
\]
By remembering these steps, Mrs. Owen can effectively use the Unit Circle to find cosine values for various angles!
1. **Understanding the Unit Circle:** The Unit Circle is a circle with a radius of 1 centered at the origin (0, 0) of a coordinate plane. The angle is measured from the positive x-axis, rotating counterclockwise for positive angles and clockwise for negative angles.
2. **Identifying the Angle:** The angle \( 300^\circ \) is greater than \( 270^\circ \) and less than \( 360^\circ \). This means it is located in the fourth quadrant of the Unit Circle.
3. **Finding the Reference Angle:** The reference angle is the acute angle formed with the x-axis. To find the reference angle for \( 300^\circ \), we can subtract it from \( 360^\circ \):
\[
360^\circ - 300^\circ = 60^\circ.
\]
Thus, the reference angle is \( 60^\circ \).
4. **Finding the Cosine Value:** In the Unit Circle, the coordinates of the angle \( 300^\circ \) correspond to \( (x, y) \), where \( x = \cos 300^\circ \) and \( y = \sin 300^\circ \). Since \( 300^\circ \) is in the fourth quadrant, the cosine value is positive.
5. **Using the Reference Angle:** The cosine of \( 300^\circ \) is the same as the cosine of its reference angle \( 60^\circ \):
\[
\cos 60^\circ = \frac{1}{2}.
\]
Since \( 300^\circ \) is in the fourth quadrant, we have:
\[
\cos 300^\circ = \cos 60^\circ = \frac{1}{2}.
\]
So, **the answer is**:
\[
\cos 300^\circ = \frac{1}{2}.
\]
By remembering these steps, Mrs. Owen can effectively use the Unit Circle to find cosine values for various angles!