To solve this problem, we need to find the quadrant, the reference angle, and the cosine ratio for the given angle, which is $300^\circ$. Here's a detailed step-by-step process:

Step 1: Determine the Quadrant

First, we determine in which quadrant the terminal side of the angle $300^\circ$ lies.

• Angles between $0^\circ$ and $90^\circ$ are in Quadrant I.
• Angles between $90^\circ$ and $180^\circ$ are in Quadrant II.
• Angles between $180^\circ$ and $270^\circ$ are in Quadrant III.
• Angles between $270^\circ$ and $360^\circ$ are in Quadrant IV.

Since $300^\circ$ is between $270^\circ$ and $360^\circ$, it lies in Quadrant IV.

Step 2: Find the Reference Angle

The reference angle is the acute angle between the terminal side of the given angle and the nearest part of the x-axis. To find the reference angle for angles in Quadrant IV, we subtract the angle from $360^\circ$:

$\text{Reference Angle} = 360^\circ - 300^\circ = 60^\circ$

So, the reference angle for $300^\circ$ is $60^\circ$.

Step 3: Determine the Cosine Value

Now, we find the value of $\cos 300^\circ$ using the reference angle. Since the reference angle is $60^\circ$, and the cosine of $60^\circ$ is a known trigonometric value, we can use that.

$\cos 60^\circ = \frac{1}{2}$

However, in Quadrant IV, the cosine function is positive (because cosine is positive for angles where the x-coordinate is positive, which is true in Quadrant IV). Therefore:

$\cos 300^\circ = + \frac{1}{2}$

Summary

• Quadrant: $300^\circ$ is in Quadrant IV.
• Reference Angle: The reference angle is $60^\circ$.
• Cosine Value: $\cos 300^\circ = \frac{1}{2}$.

Next Steps or Questions?

Would you like further details on any of these steps? If yes, feel free to ask!

Here are 5 questions to expand on the concept:

1. What is the sine of $300^\circ$?
2. How would you find the cosine of an angle in Quadrant III?
3. What is the reference angle for $135^\circ$?
4. Can you explain how to determine the sign of the sine in different quadrants?
5. What is the general formula to find the reference angle for any given angle?

Tip:

When working with trigonometric functions, always keep track of the sign of the function in the quadrant where the angle lies!