Finding Reference Angle for Angles not on the Unit Circle

To find the reference angle for an angle not on the unit circle, we first need to understand the definition of a reference angle. A reference angle is the acute angle formed by the terminal side of the given angle and the horizontal axis. The reference angle is always between 0° and 90°, and it is always positive.

Let’s consider an example: Find the reference angle for the angle 250°.

Step 1: Determine the quadrant in which the given angle lies. Since 250° is between 180° and 270°, it lies in the third quadrant.

Step 2: Use the formula for the reference angle in the third quadrant:

$ \theta_{reference} = \theta – 180° $

For our example:

$ \theta_{reference} = 250° – 180° = 70° $

Therefore, the reference angle for 250° is 70°.

Given an angle not on the unit circle, say 315°, we need to find its reference angle.

Step 1: Identify the quadrant of the angle. Since 315° is between 270° and 360°, it is in the fourth quadrant.

Step 2: The formula for the reference angle in the fourth quadrant is:

$ heta_{reference} = 360° – heta $

Applying this formula:

$ heta_{reference} = 360° – 315° = 45° $

Thus, the reference angle for 315° is 45°.

Find the reference angle for the angle 140°.

Since 140° is in the second quadrant:

$ heta_{reference} = 180° – heta $

So:

$ heta_{reference} = 180° – 140° = 40° $

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